Showing papers on "Field (mathematics) published in 1972"

Automorphisms and derivations of incidence algebras

Abstract: This paper studies the derivations and automnorphisms of the incidence algebra of a locally finite partially ordered set. Two subspaces are shown to span the space of derivations: the space of inner derivations and the space of derivations associated with the additive functions. An analogous result is shown for the group of automorphisms. A number of dimension calculations are also made. 1. Preliminaries. In the study of additive and multiplicative functions on the segments of a (partially) ordered set, it is natural to consider derivations and automorphisms, respectively, of the incidence algebra. In ?2, we study the derivations. In ?3, we give the analogous results for automorphisms. For a more complete description of the notation used see [1]. Let P be a locally finite (partially) ordered set, i.e. for which every segment [x,y]={z|x

Algebraic Structure of Linear Dynamical Systems. III. Realization Theory Over a Commutative Ring

Abstract: The realization theory linear dynamical systems, previously developed over a field, are extended to a large class of commutative rings. The principal result is that the existence criterion for a finite realization extends without modification from a field to a Noetherian integral domain.

Elementary statements over large algebraic fields

Abstract: We prove here the following theorems: A. If k is a denumerable Hilbertian field then for almost all (al. ae) 6 9(k,/k)e the fixed field of {al,. . ., ae}, k,(al. . ., ae), has the following property: For any nonvoid absolutely irreducible variety V defined over ks(ai, ae) the set of points of V rational over K is not empty. B. If E is an elementary statement about fields then the measure of the set of a E 9(Q/Q) (Q is the field of rational numbers) for which E holds in Q(a) is equal to the Dirichlet density of the set of primes p for which E holds in the field F, of p elements. Introduction. Denote by E the class of all fields K which have the following property: For any nonvoid absolutely irreducible variety V defined over K, the set of points of V rational over K is not empty. For any prime p denote by Fp the field with p elements. Then it follows from the Riemann hypothesis for curves that if =37=J Fp/D is a nonprincipal ultra-product of the Fp then Y E E (see [1, Theorem 6]). On the other hand, it follows from the Hilbert Nullstellensatz that if K is an algebraically closed field then K e E. In particular it follows that the algebraic closure of Q (the field of rational numbers), Q, belongs to E. It is therefore natural to ask whether or not J# r) Q ES. Ax gave a counterexample in [2, ?14], showing that this is not always the case. One can then ask whether Ax's example is exceptional and that, in general, r) Q does belong to E. To be more precise denote by Q(g) the fixed field in Q of an automorphism a E 'AO/Q) (6?/(!/Q) is the Galois group of Q over Q). Ax showed [1, Theorem 5] that for every nonprincipal ultra-product Y of the Fp there exists g E 6N(&/Q) such that 5 r Q= =(g), and conversely, for each a E 'W(Q/Q) there exists a nonprincipal ultra-product Y of the Fp such that r = Q((). Furstenberg suggested to me to prove that, for almost all g E W(0/Q) (in the sense of Haar measure), Q(u) c E. More generally, let k be any field. Denote by I-'k the normalized Haar measure Received by the editors June 18, 1970. AMS 1969 subject classifications. Primary 1440, 1245.

Computational problems, methods, and results in algebraic number theory

Abstract: Finite fields.- Factorization of polynomials.- Galois groups.- Continued fractions.- Field extensions.- Modules and orders.- Products of linear forms.- Units in algebraic number fields.- Class numbers of algebraic number fields.- Class groups and class fields of algebraic number fields.- Diophantine equations.- The hasse principle for cubic surfaces.

Representations of Chevalley groups in characteristic $p$

Abstract: If GK is a Chevalley group over a field K of prime characteristic p, the irreducible representations of GK over K form a natural object of study. The basic results have been obtained by Steinberg [15], who showed that, if K is perfect, then each irreducible rational representation of GK over K is a tensor product of representations obtained from certain basic representations by composing them with field automorphisms. These basic representations were obtained by “integrating” the irreducible restricted representations of a restricted Lie algebra associated with the group, which had been studied earlier by Curtis [7]. The present author had obtained the main results previously for the groups SL(n, K), Sp(2n, K) by different means, involving reduction (mod p) from the characteristic 0 case [16]. In this paper we extend this method to the other types of groups, in the hope that some additional insight may be gained.

Essays in algebraic simplification

Abstract: This thesis consists of essays on several aspects of the problem of algebraic simplification by computer Since simplification is at the core of most algebraic manipulations, efficient and effective simplification procedures are essential to building useful computer systems for non-numerical mathematics Efficiency is attained through carefully designed and engineered algorithms, heuristics, and data types, while effectiveness is assured through theoretical considerations Chapter 1 is an introduction to the field of algebraic manipulation, and serves to place the following chapters in perspective Chapter 2 reports on an original design for, and programming implementation of, a pattern matching system intended to recognize non-obvious occurrences of patterns within algebraic expressions A user of such a system can "teach" the computer new simplification rules Chapter 3 reports on new applications of standard mathematical algorithms used for canonical simplifications of rational expressions These applications, in combinations, allow a computer system to contain a fair amount of expertise in several areas of algebraic manipulation Chapter 4 reports on a new, practical, canonical simplification algorithm for radical expressions (ie algebraic expressions including roots of polynomials) The effectiveness of the procedure is assured through proofs of appropriate properties of these simplified expressions Chapter 5 is a brief summary and a discussion of potential research areas Two appendices describe MACSYMA, a computer system for symbolic manipulation, an effort of some dozen researchers (including the author) which has served as the vehicle for this work

Garding domains and analytic vectors for quantum fields.

Abstract: If one studies the canonical commutation relations (CCR's) of quantum field theory in the unitary Weyl form, one does not know if one can find a common dense domain for the field operators since their domain of definition depends on the test function. We consider here a general class of test function spaces including the spaces S and D of Schwartz and the space U0≃R(∞) of all finite linear combinations of a countable basis. It is shown that there exists an invariant Garding domain D on which all fields are defined and strongly continuous. D consists of analytic vectors for the fields. It turns out that the test function space can be enlarged by continuity. For irreducible or factor representations it becomes even a Hilbert space. The basic idea of the proof is the same as in the Schrodinger representation for one degree of freedom and very transparent. We simply use rapidly decreasing functions in ``Q‐space'' and ``P‐space'' as smoothing factors. That this can be done in the infinite case also is due to a...

Formulas for some diophantine approximation constants

Abstract: Let M be a full Z-module in F a real number field of degree at least 3 with N(a) denoting the norm of a e F. Given any nonzero number in M we make the plausible conjecture that one can find a number 0 in M such that N(0) = N(tj>) and the algebraic conjugates of 0 (not including 0) have ratios arbitrarily near any given numbers consistent with the complex algebraic conjugates of elements of F. We use the conjecture to give explicit formulas for some diophantine approximation constants. Without the conjecture our methods lead to corresponding lower bounds for these constants.

The Resultant and the Matrix Equation $AX = XB$

Abstract: The concept of a resultant of two polynomials over an arbitrary field is used to reconcile the Cecioni–Frobenius theorem with the theory of direct products, insofar that these are used to establish the existence of a nonzero solution to the matrix equation $AX = XB$.

Generalized rational identities

Abstract: Publisher Summary This chapter discusses the generalized rational identities It presents a simple proof of the Amitsur–Bergman result with a somewhat different condition on the centers (which are still assumed infinite) The main tool is the notion of a universal skew field of fractions This was constructed by Amitsur in the special case of free algebras, using precisely his results on rational identities The explicit construction for universal skew fields of fractions to prove the result on rational identities is used If R is any ring, then by afield of fractions of R, one understands a field K with an embedding R → K such that K is the field generated by the image In the commutative case, such a K exists if R is an integral domain, and it is then unique In general no necessary and sufficient conditions are known for a field of fractions to exist and even when it does exist, it need not be unique

Transformations of symmetric tensors

Abstract: This paper is about linear transformations of the fc-fold symmetric tensor product of an ^-dimensional vector space V which carry nonzero decomposable tensors to nonzero decomposable tensors The main theorem shows that every such transformation is induced by a nonsingular transformation of V provided both (i) the field has characteristic either 0 or a prime greater than k and every polynomial over the field with degree at n is a product of linear factors (ii) n>k + l

On elements with negative squares

Abstract: We prove that in a partially ordered linear algebra no element can have a square which is the negative of an order unit. In particular, the square of a real matrix cannot consist entirely of negative entries. We generalize the well-known theorem that the complex numbers admit no lattice order. It is a simple matter, using the Perron-Frobenius theorem [3], to show that the square of a real (finite) matrix cannot consist entirely of negative entries. In this paper we give an alternate proof of this result which makes use only of some elementary order properties. It is appropriate, therefore, to construct a proof within the more general framework of partially ordered linear algebras. The main theorem will then be valid not only for finite matrices but also for (row-finite) infinite matrices as well as operators on a Banach space [1]. We also generalize a result due to Birkhoff and Pierce [2] which states that the complex number field regarded as a linear algebra over the reals admits no lattice order. DEFINITION. A partially ordered linear algebra (p.o.l.a.) P is a real associative linear algebra on which there is defined a partial ordering which satisfies the following conditions (x, y, z denote elements of P and a. denotes a real number): (a) ifx 0 and z > 0 such that x = y z. We also make use of the following condition: (d*) for any x there exists a y such that -y < x _ y. It is easy to verify that conditions (a), (b), (c), and (d) are equivalent to (a), (b), (c), and (d*). It can be shown, however, that a stronger set of conditions is obtained if (d) is replaced by the condition that the partial ordering be a lattice order. We make use of the standard elementary Received by the editors February 17, 1971. AMS 1970 subject classiflcations. Primary 06A70; Secondary 15A45, 47B55.

Double commutants of algebraic operators

Abstract: The double commutant of an algebraic operator on a complex Hilbert space is equal to the algebra (with identity) generated by that operator. Introduction. Let fl be an algebra over some field, and let S? be a subset of t. The commutant of 5 in W is the set S?' of all elements of Qi which commute with every element of Yf. The double commutant of Y? in W is the commutant of Y?' in W. If 9Y consists of a single element s, we shall speak of the commutant and double commutant of s rather than of {s}. Let A be a linear transformation on a vector space. We say that A is algebraic if there exists a polynomial p such that p(A) = 0. In this paper we shall prove that if X' is a complex Hilbert space, and A belonging to B(Q#) (the algebra of all bounded linear' transformations on JX) is algebraic, then the double commutant of A in B(,#') is equal to the algebra generated by A. This generalizes the well-known analogous result for finite-dimensional spaces (see [2, p. 113]), since any linear transformation on a finitedimensional vector space is algebraic. It is already known (see [3 p. 72]) that if T is an algebraic linear transformation on a vector space I then the double commutant of T in L('f) (the algebra of all linear transformations on f) is equal to the algebra generated by T. This result does not imply our result for B(JV') however, since in general L(X#) is far larger than B(Q#). In the forthcoming the word operator will mean bounded linear transformation, and Hilbert space will mean complex Hilbert space. 1. Notation. If A is an algebraic operator on a Hilbert space JX', we denote by WA the algebra (with identity) generated by A in B(J). REMARK. In discussing operators on Hilbert space one often speaks of the weakly closed algebras which they generate. Since the algebra generated by an algebraic operator is finite dimensional it is closed in all of the usual Received by the editors September 29, 1971. AMS 1970 subject classifications. Primary 46L15, 47B99.

Constructing sequences of divided powers. II

Abstract: In my Sequences of divided powers in irreducible, cocommutative Hopf algebras, I demonstrated the existence of extensions of sequences of divided powers over arbitrary fields, if certain coheight conditions are met. Here, I show that if the characteristic of the field does not divide n, every sequence of divided powers of length n 1, in a cocommutative Hopf algebra, has an extension that can be written as a polynomial in the previous terms. (An algorithm for finding these polynomials is given, together with a list of some of them.) Furthermore, I show that if one uses this method successively for constructing a sequence of divided powers over a primitive, the only obstructions will occur at powers of the characteristic of the field. Some of the basic definitions of this paper are the following: (1) If H is a Hopf algebra and 0 $ g E H, then g is a grouplike if Ag = g 0 g. (2) If h c H, then h is a primitive if Ah = h 0 1 + 1 ? h. (3) A Hopf algebra will be called irreducible if every nontrivial subcoalgebra contains a fixed, nontrivial subcoalgebra, i.e., if H is irreducible, the identity is the unique grouplike. (4) An irreducible Hopf algebra will be called graded, if there exists a set of subspaces {Hj}j? O such that (a) H= G`0H,; (b) Ho= 1 k; (c) AHiC E=o Hj? Hi-; (d) HiHjC Hi+j (5) A sequence of divided powers Ox = 1, 1x, 2x, , x is a set of elements in a cocommutative, irreducible Hopf algebra such that Altx = 0 o t-iX, for all 0 < t < n. Received by the editors December 8, 1970. AMS 1970 subject classiflcations. Primary 16A24.

Balanced and $QF-1$ algebras

Abstract: A ring R is QF-1 if every faithful module has the double centralizer property. It is proved that a local finite dimensional algebra is QF-1 if and only if it is QF. From this it follows that an arbitrary finite dimensional algebra has the property that every homomorphic image is QF-1 if and only if every homomorphic image is QF. Throughout the following all rings are associative and have identity, all modules are unitary and all algebras are finite dimensional over a field. If M is a module over a ring R, we write End(M) to represent its set of R-endomorphisms viewed as a ring of operators on the opposite side of M. Then M is an End(M)-R bimodule, and calling its ring of End(M) endomorphisms Bi End(M), there is a natural ring homomorphism R->-Bi End(M) via rF-+multiplication by r. If this ring homomorphism is surjective the module M is said to be balanced, or to have the double centralizer property. After proving, with C. J. Nesbitt, that every faithful module over a quasi-Frobenius (QF) algebra is balanced [151 (a fact now well known for QF rings (see, for example, [2, ?59])), R. M. Thrall gave an example of a ring over which every faithful module has the double centralizer property which is not QF [16]. He called the above rings QF-1 rings and posed the problem of characterizing them in terms of their ideal structure. Thrall's problem is still unsolved, even for algebras, though various partial results may be found in [1], [3], [5], [6], [10]. Here, we offer a solution to a modification of Thrall's problem posed in [1] and [3]. A ring is balanced in case all of its modules (faithful or not) are balanced. It is our intention to prove that an algebra is balanced if and only if it is in fact a uniserial algebra in the sense of Koethe and Nakayama [13]. Thus we verify, for algebras, a recent conjecture of J. P. Jans [8] (cf. [7, Remark (d)]) and extend his theorem that a balanced Presented to the Society, November 23, 1970; received by the editors December 5,

ON THE ALGEBRAIC CLOSURE OF A LOCAL FIELD FOR p = 2

Abstract: Let k be a finite extension of the field of 2-adic numbers. In this paper we determine the structure of the Galois groups of the algebraic closure and of the maximal extension without simple ramification of the field k under the assumption that the maximal extension without higher ramification of the field k contains a fourth root of 1.

A characterization of general Z.P.I.-rings. II

Abstract: A commutative ring A is a general Z.P.I.-ring if each ideal of A can be represented as a finite product of prime ideals. We prove that a commutative ring A is a general Z.P.I.-ring if each finitely generated ideal of A can be represented as a finite product of prime ideals. We also give a characterization of Krull domains in terms of *-operations, as defined by Gilmer. Introduction. A commutative ring A is a iT-ring if each principal ideal of A can be represented as a product of prime ideals; A is a p-ring if each finitely generated ideal of A can be represented as a product of prime ideals. A commutative ring A is a general Z.P.I.-ring if each ideal of A can be represented as a product of prime ideals. Mori characterized nr-rings in [4], [5] and general Z.P.I.-rings in [6]. In [9] Wood extended Mori's results on nr-rings. Wood also gave a characterization of general Z.P.I.rings which is independent of Mori's work [10]. Let D be an integral domain with identity and with quotient field K. If F(D) is the set of nonzero fractional ideals of A, a mapping B--*B* of F(D) into F(D) is called a *-operation on D if the following three conditions hold for any a in K-{O}, and any B, C in F(D). (1) (a)*=(a), (aB)*=aB*. (2) BiB*, if Bc C, B*c C*. (3) (B*)*=B*. If (B)* (C)*, B and C are called *-equivalent, denoted B,,* C. If B=B*, B is called a *-ideal. An integral domain D is a Krull domain if D= n v, where { V} is a set of rank one discrete valuation rings with the property that for each nonzero x in D, xVa= Va for all but a finite number of the Va. In the first section of this paper, we give a new characterization of Krull domains. We prove that an integral domain A with identity is a Krull domain if and only if there is a *-operation on A such that each nonzero principal ideal of A is *-equivalent to a product of prime ideals. Then we consider nr-rings with no zero divisors, called ,r-domains. Theorem 1.2 states that A is a nr-domain with identity if and only if A is a Krull domain Received by the editors June 28, 1971. AMS 1970 subject class/ifcations. Primary 13F05, 13F10.

On a semi-direct product decomposition of affine groups over a field of characteristic $0$

Abstract: PROOF. The Jacobson radical of a ring is the intersection of its maximal left (or right) ideals. Dually the coradical of a coalgebra C over a field is identical with the socle of C as a right (or left) C-comodule. Since k is perfect, the coradical of fc(g)fc A is k®kR, where k is the algebraic closure of k. Hence we can assume that k — k. Moreover A can be assumed to be finitely generated as a fc-algebra. Let Vi9 i = 1, 2 be two finite dimensional right A-comodules. Becauce G(A°) = Algfc(̂ 4., k) is dense in A* = Hom^A, k) [6, Lem. 3.6], V* is a semisimple A-comodule iff Vi is a semisimple left G(A°)-module. Hence by the remark above if V4 are semisimple, then Vι ® V2 is also semi-simple. This means that R (x) R is a semisimple right A-comodule. Since the multiplication μ: A (x) A —> A is a right A-comodule map, R R is contained in R. Clearly R is stable under the antipode of A. Hence R is a sub-Hopf algebra of A.

The size function of abelian varieties

Abstract: The size function is defined for points in projective space over any field K, finitely generated field over Q, generalizing the height function for number fields. We prove that the size function on the K-rational points of an abelian variety is bounded by a quadratic function. Introduction. In his book, Introduction to transcendental numbers, Lang showed how one can extend some of the theorems about the exponential function ex to theorems about the exponential map from complex g-space to the complex points of group varieties of dimension g, defined over the complex numbers. Looking at transcendental numbers in this general setting, he raised an arithmetic-geometric question about the addition formula of a group variety. In this paper, we shall answer this question in the case of an abelian variety. In his report to Seminaire Bourbaki in May 1964, [6], Lang described the following result of Neron and Tate: If A is an abelian variety defined over a number field K, there exist a quadratic function Q and a linear function L from A(K), the Krational points of A, to the real numbers, such that the logarithmic height function, h: A(K) -R, defined with respect to any closed immersion in projective space, is additively equivalent to the function Q+L. Our main result, Theorem 3.5, is a generalization of this (albeit in a weaker form), to the size function, which is defined for an abelian variety defined over any field of characteristic 0. It states that there is a quadratic function Q: A(K) -> R such that size(x) < Q(x) for all x E A(K). I wish to take this opportunity to thank Professor Serge Lang who introduced me to the problem and who helped and encouraged me in my work. This work was partially supported by a National Science Foundation Graduate Fellowship. 1. Let K be a field which is finitely generated over Q. K has a proper set of generators {t,, . . ., tr, u} over Q, denoted {t, u}, where proper means that {t,, . . ., tr} is a transcendence base of K over Q and u is integral over Z [tl,. . ., tr]. Let q = [K: Q(t)]. An element a E K is said to be an integral coordinate with respect to {t, u} if, when a is expressed as a linear combination of {1, u,. .., u q-l} with coefficients in Q(t) in lowest terms, all coefficients lie in Z [t ]. Note that if a, f E K are integral coordinates with respect to {t, u}, then the sum a + and the product afi are integral coordinates with respect to {t, u}. Received by the editors November 10, 1970. AMS 1969 subject classifications. Primary 1032, 1275, 1440, 1450, 1451.

FIELD THEORY WHERE ALL GAMMA/sub jk//sup i/ INVARIANTS VANSIH.

Abstract: We investigate whether a nontrivial field theory exists for which all scalars formed fromΓ jk i (whereg ij is used to raise and lower indices) can be zero for all points of space and time. We find some examples for which the invariants are all zero for all points at which the field is finite. We also comment upon the problem of boundary conditions, in general.

On subfields of countable codimension

Abstract: In [2] the authors asked if any two real closed subfields R, R' of the field of complex numbers C such that R(x/ (-1))R'(V(1)) =C are isomorphic. It is not difficult to see that the answer is negative. This is proved in the first part of the note. In the second we study the problem if any field which is not prime contains a proper subfield of countable (finite or infinite) codimension. 1. PROPOSITION 1. Let R be any real closed field of power continuum. Then the field R(V/(1)) is isomorphic to the field of complex numbers. PROOF. It is well known (see e.g. [3, p. 274]) that if the field R is real closed then the field R(V/(1)) is algebraically closed. Since any algebraically closed field of power continuum and of characteristic zero is isomorphic to the field of complex numbers hence if R is real closed of power continuum then R(J(1)) is isomorphic to the field of complex numbers. Thus the proposition is proved. Because there exist nonisomorphic real closed fields of power continuum (e.g. the field R and the real closure of the ordered field R(t), where 0< t

The Schur index and roots of unity

Abstract: A short proof is given for the main step in the proof of the theorem of Benard and Schacher which asserts that if the Schur index of a character X of a finite group is m then the mth roots of unity lie in the field of values Q(x). One of tke most elegant results about the Schur index of a character of a finite group is the recent one by M. Benard and M. Schacher. They prove the Schur index over the rationals of the character x can be m only when the field of values of X contains the mth roots of unity. Their proof uses the Witt-Brauer reduction to the case of a "special" character, the theory of Hasse invariants and a rather long computational theorem of C. Ford [1]. We give here a proof of Ford's result (including a case not covered by him) in a version which covers the bulk of the BenardSchacher result. The proof uses only the most elementary facts about crossed products and the Brauer group. In order to make the proof even shorter, we shall state it as a result about algebras. For the passage from characters to special characters and then to algebras, the reader may consult any of the references listed. Let K be a field. A cyclotomic algebra over K is a crossed product A\=(K(E), G, O=2aG K(e)ur in which E is a root of unity, G is the Galois group of K(E) over K, and # is a factor set whose values are roots of unity in K(E). The multiplication in A is given by the rules uax = r(x)u,a u Uatur = l(AC, T)uar for all x in K(E) and o, X in G. THEOREM. If the cyclotomic algebra A has exponent m in the Brauer group of K, then the mth roots of unity lie in the center K. PROOF. We shall change factor sets in the proof so we write A(#) for the crossed product made with K(E), G and jl. Let the values of # generate a group (Kc) of nth roots of unity. In the Brauer group of K, we have [A (O] = [A(V"A)] = [A(1)] = 1. Received by the editors February 7, 1972. AMS 1970 subject classifications. Primary 13A20, 20C15.

Grothendieck and Witt rings of hermitian forms over Dedekind rings

Abstract: The prime ideal theory of the Grothendieck and Witt ring of non-degenerat e hermitian forms over a Dedekind ring with involution is studied. The relationship of these rings to those defined over the quotient field of the Dedekind ring is also investigated. The main goal of this paper is to extend the structure theory for Witt rings over fields of Poster [18] and Harrison-Leicht-Lorenz ([10], [16]) to the Grothendieck ring K(C, J) and the Witt ring W(C, J) of a Dedkind ring C with an involution J. Since the case J = identity is allowed, the Grothendieck and Witt rings of [12] are included. We shall see that the main theorems of Pfister and Harrison-LeichtLorenz remain true for W(C, J) and that if J is the identity they are also true for K(C, J). However, for K(C, J) with J ^ identity there is some deviation: there may be p-torsion for primes p 2 and there may be nilpotent elements which are not torsion (Example 1.3). This fact has been overlooked in [13]. In §1 we extend some elementary results of [12, §11, §13] to the case J ^ identity. We conjecture that they are well known to the specialists but we did not find an appropriate reference in the literature. We show that the canonical map from W(C, J) to the Witt ring W(L, J) of the quotient field L of C is infective and give some information about the kernel A(C, J) of the map K(C, J) —> K(L, J). Since the exact determination of A(C, J) is not needed for our structure theory we delay this matter to §4, where such a deter­ mination is given along the same lines as in [12, §11.2], We then show that W(C, J) is the intersection of certain subrings W(C„ J) of W(L, J) which are Witt rings for abelian groups of exponent 2 in the sense of [14, Def. 3.12] and we describe the image K'(Cf J) of K{C, J) in K(L, J) in an analogous way. We are thus led to study subrings T of an "abstract" Witt ring R for an arbitrary abelian g-group [14, Def. 3.12]. If T is the inter­ section of a family {Ta} of subrings of R which are also Witt rings for some abelian g-groups, the entire prime ideal theory of R remains true for T. In § 3 we show that if T is either K(C, J) or W(C, J) then the group of units of T is generated by 1 + Nil T and the rank one spaces