Showing papers on "Ring (mathematics) published in 1974"
TL;DR: The first of a series of papers dealing with the representation theory of artin algebras is presented in this paper, where the main purpose is to develop terminology and background material which will be used in the rest of the papers in the series.
Abstract: This is the first of a series of papers dealing with the representation theory of artin algebras, where by an artin algebra we mean an artin ring having the property that its center is an artin ring and λ is a finitely generated module over its center. The over all purpose of this paper is to develop terminology and background material which will be used in the rest of the papers in the series. While it is undoubtedly true that much of this material can be found in the literature or easily deduced from results already in the literature, the particular development presented here appears to be new and is especially well suited as a foundation for the papers to come.
1,267 citations
TL;DR: In this paper, it was shown that if a ring R has no nonzero nilpotent elements, satisfies a polynomial identity and has a Baer ring as center, then R must be a P.P.-ring.
Abstract: Baer rings are rings in which the left (right) annihilator of each subset is generated by an idempotent [6]. Closely related to Baer rings are left P.P.-rings; these are rings in which each principal left ideal is projective, or equivalently, rings in which the left annihilator of each element is generated by an idempotent. Both Baer and P.P.-rings have been extensively studied (e.g. [2], [1], [3], [7]) and it is known that both of these properties are not stable relative to the formation of polynomial rings [5]. However we will show that if a ring R has no nonzero nilpotent elements then R[X] is a Baer or P.P.-ring if and only if R is a Baer or P.P.-ring. This generalizes a result of S. Jondrup [5] who proved stability for commutative P.P.-rings via localizations – a technique which is, of course, not available to us. We also consider the converse to the well-known result that the center of a Baer ring is a Baer ring [6] and show that if R has no nonzero nilpotent elements, satisfies a polynomial identity and has a Baer ring as center, then R must be a Baer ring. We include examples to illustrate that all the hypotheses are needed.
379 citations
01 Jan 1974
TL;DR: In the first part of this chapter, a general structure theory for rings is presented as mentioned in this paper, and the reader would do well to read both these discussions before beginning serious study of the chapter.
Abstract: In the first part of this chapter a general structure theory for rings is presented. Although the concepts and techniques introduced have widespread application, complete structure theorems are available only for certain classes of rings. The basic method for determining such a class of rings might be described intuitively as follows. One singles out an “undesirable” property P that satisfies certain conditions, in particular, that every ring has an ideal which is maximal with respect to having property P. This ideal is called the P-radical of the ring. One then attempts to find structure theorems for the class of rings with zero P-radical. Frequently one must include additional hypotheses (such as appropriate chain conditions) in order to obtain really strong structure theorems. These ideas are discussed in full detail in the introductions to Sections 1 and 2 below. The reader would do well to read both these discussions before beginning serious study of the chapter.
317 citations
Patent•
02 Dec 1974TL;DR: In this article, a ring protection hardware system is used to prevent processes from intering with each other or sharing each other's address space in an unauthorized manner in hardware/firmware by restricting addressability to a segmented memory.
Abstract: Computer data and procedure protection by preventing processes from intering with each other or sharing each other's address space in an unauthorized manner is accomplished in hardware/firmware by restricting addressability to a segmented memory and by a ring protection mechanism. To protect information in segments shared by several processes from misuse by one of these processes a ring protection hardware system is utilized. There are four ring classes numbered 0 through 3. Each ring represents a level of system privilege with level 0 (the innermost ring) having the most privilege and level 3 (the outermost ring) the least. Every procedure in the system has a minimum and a maximum execute ring number assigned to it which specifies who may legally call the procedure. Also maximum write and read ring numbers specify the maximum ring numbers for which a write and/or read operation is permitted. Processes use a segmented address during execution wherein segment tables isolate the address space of the various processes in the system. Hardware checks that the address used by a process is part of the address space assigned to the process, and if the address is outside the prescribed address space, an exception occurs. A process cannot refer to data within the address space of another process because the hardware uses the segment table of the referencing process.
120 citations
TL;DR: In this article, it was shown that R is an elementary divisor ring if and only if every finitely presented module over R is a direct sum of cyclic modules, thus providing a converse to a theorem of Kaplansky and answering a question of Warfield.
Abstract: Throughout, rings are commutative with unit and modules are unital. We prove that R is an elementary divisor ring if and only if every finitely presented module over R is a direct sum of cyclic modules, thus providing a converse to a theorem of Kaplansky and answering a question of Warfield. We show that every Bezout ring with a finite number of minimal prime ideals is Hermite. So, in particular, semilocal Bezout rings are Hermite answering affirmatively a question of Henriksen. We show that every semihereditary Bezout ring is Hermite. Semilocal adequate rings are characterized and a partial converse to a theorem of Henriksen is established.
94 citations
TL;DR: The main idea is to exploit a result in [17], which says that local rings of small embedding codimension and depth ≥ 3 are parafactorial, and tell us, with suitable additional hypotheses, that the ring is factorial, or Gorenstein, or even a complete intersection.
Abstract: Let A be a local ring of dimension d. If A is a quotient of a regular local ring of dimension n = d+r, then we say that A has embedding codimension ≤ r. This paper investigates some special properties of local rings of small embedding codimension. The main idea is to exploit a result in [17], which says that local rings of small embedding codimension and depth ≥ 3 are parafactorial. This tells us, with suitable additional hypotheses, that the ring is factorial, or Gorenstein, or even a complete intersection.
83 citations
71 citations
TL;DR: In this article, it was shown that if G = O(t, K), the orthogonal group, and K a field of characteristic zero, then RG is Cohen-Macaulay for an appropriate action of G.
Abstract: Theorem. Let R be a commutative Noetherian ring with identity. Let M = (ci1) be an s by s symmetric matrix with entries in R. Let I the be ideal of t + 1 by t+ 1 minors of M. Suppose that the grade of I is as large as possible, namely, gr I = g = s(s + 1)/2 st + t(t1)/2. Then I is a perfect ideal, so that RI is Cohen Macaulay if R is. Let G be a linear algebraic group acting rationally on R = K[xl, . . . X,]. Hochster has conjectured that if G is reductive, then RG is Cohen-Macaulay, where RG denotes the ring of invariants of the action of G. The above theorem provides a special case of this conjecture. For G = O(t, K), the orthogonal group, and K a field of characteristic zero, the above yields: Corollary. For R and G as above, RG is Cohen-Macaulay for an appropriate action of G. In order to obtain these results it was necessary to prove a more general form of the theorem stated above, which in turn yields a more general form of the corollary. 0. Introduction. Let G be a linear algebraic group acting rationally on an ndimensional K-vector space (see [2, p. 94]). We may consider G as acting on the 1-forms of R = K[xl...,xj]; the action then extends uniquely to R. Hochster has conjectured that if G is reductive, then the ring of invariants RG is CohenMacaulay [10]. In particular, the hypothesis holds if G is one of the classical groups. The conjecture is known in the following cases (see [10], [11]): (i) G = GL(t) acts on K(r+s)t via: X, Y are r by t and t by s matrices of indeterminates, respectively, and A E G acts by taking entries of X to those of XA-I and entries of Y to those of A Y. (ii) G = SL(t) and acts on a t by r matrix of indeterminates X by taking X to AXforallA e G. (iii) G = GL(1)m, the m-torus, and the representation is arbitrary. (iv) Various representations of products of SL(t) with several copies of GL(m), m varying. (v) G is finite and the representation is arbitrary. This paper establishes the result for the case where G = 0(t), the orthogonal group, and the representation is as in (ii) above. In this case RG = K [entries of XtX] = K[ UL4]/Q, where (UU) is a symmetric matrix of indeterminates Received by the editors March 5, 1973. AMS (MOS) subject classqiBcations (1970). Primary 13C05, 13C10; Secondary 15A72.
66 citations
TL;DR: In this article, it was shown that the Gabriel dimension is inherited by a polynomial ring, and upper and lower bounds for this dimension are obtained for the class of modules with Krull dimension.
60 citations
01 Mar 1974
TL;DR: In this paper, it was shown that if A contains no non-zero nilpotent element, then A is regular iff every principal left ideal is the left annihilator of an element of A.
Abstract: Throughout, A denotes an associative ring with identity and “module” means “left, unitary A-module”. In (3), it is proved that A is semi-simple, Artinian if A is a semi-prime ring such that every left ideal is a left annihilator. A natural question is whether a similar result holds for a (von Neumann) regular ring. The first proposition of this short note is that if A contains no non-zero nilpotent element, then A is regular iff every principal left ideal is the left annihilator of an element of A. It is well-known that a commutative ring is regular iff every simple module is injective (I. Kaplansky, see (2, p. 130)). The second proposition here is a partial generalisation of that result.
58 citations
TL;DR: In this paper, the authors introduce a wider class of ring groups, one including the locally compact groups, and a construction is given whereby to each ring group there is defined a dual ring group.
Abstract: A number of authors have introduced ring groups as objects generalizing locally compact groups. An analogue of the Pontrjagin principle of duality holds for ring groups. In this paper we introduce a wider class of ring groups, one including the locally compact groups.A construction is given whereby to each ring group there is defined a dual ring group ; here . By definition a ring group is determined by a -algebra (the space of the ring group) equipped with an additional structure which allows to be considered, in particular, as a Hopf-von Neumann algebra. When is a locally compact group, is the -algebra of bounded measurable functions on , considered in the natural way as operators in .Bibliography: 15 items.
TL;DR: In this article, the algebraic structure of linear systems defined over R[λ], the ring of polynomials in λ with real coefficients, is studied and natural definitions of controllability and observability are introduced.
TL;DR: In this article, a finite PI ring which cannot be embedded in matrices over any commutative ring was constructed, including a semiprime PI ring with no classical ring of quotients, showing that the property of having all regular elements invertible is not inherited by matrix rings.
Abstract: Several examples are constructed, including a finite ring which cannot be embedded in matrices over any commutative ring, a semiprime PI ring with no classical ring of quotients, an example showing that the property of having all regular elements invertible is not inherited by matrix ringsMn(R), and a prime PI ringR with an idempotente such thatR/ReR has finitely generated projective modules not induced by any finitely-generated projective R-module.
TL;DR: In this article, it was shown that if A = (m in) C Mn(R) is a tiled R-order of finite global dimension, then Ai, < n 1 for all i, j.
Abstract: Let R be a discrete valuation ring with maximal ideal m and the quotient field K Let A = (mA"j) C M_(K) be a tiled R-order, where Xii e Z and Ai = 0 for 1 : i s,n The following results are proved Theorem 1 There are, up to conjugation, -only finitely many tiled R-orders in M,(K) of finite global dimension Theorem 2 Tiled R-orders in M"(K) of finite global dimension satisfy DCC Theorem 3 Let A C Mn(R) and let r be obtained from A by replacing the entries above the main diagonal by arbitrary entries from R If r is a ring and if gl dim A 0 for all i, j (cf Lemma 11) One of the main results in this paper shows that if A = (m in) C Mn(R) is a tiled R-order of finite global dimension, then Ai, < n 1 for all i, j; hence it follows that there are only finitely many tiled R-orders in M"(R) of finite global dimension Using this we show that if S1, S2, , Sh is a finite family of Presented to the Society, January 12, 1972; received by the editors June 15, 1973 AMS(MOS) subject classifications (1970) Primary 16A60
TL;DR: For manifolds with actions of compact Lie groups, a homomorphism from the bordism ring of -manifolds to the ring of rational Hirzebruch genus is constructed in this paper.
Abstract: In this article, manifolds with actions of compact Lie groups are considered. For each rational Hirzebruch genus , an equivariant genus , a homomorphism from the bordism ring of -manifolds to the ring , is constructed. With the aid of the language of formal groups, for some genera it is proved that for a connected compact Lie group , the image of belongs to the subring . As a consequence, extremely simple relations between the values of these genera on bordism classes of -manifolds and submanifolds of its fixed points are found. In particular, a new proof of the Atiyah-Hirzebruch formula is obtained.Bibliography: 10 items.
TL;DR: In this article, the simple direct summands of the group algebra Q[G] when G = SL(2,q) are computed from the character table of G and the dimension over Q of a simple component is determined from the characters corresponding to it.
Abstract: §1. Introduction. The object of this paper is the calculation of the simple direct summands of the group algebra Q[G] when G = SL(2,q). We assume the character table of G is available and make the computation from that information. There is a well known procedure for finding the number of simple components. For each irreducible complex character γ, one forms the sum γ + γτ + … of all the algebraic conjugates of γ. The sums obtained this way correspond one-to-one with the simple components. The dimension over Q of a simple component is determined from the characters corresponding to it. Furthermore, it is a full matrix ring over a division ring. Aside from the information obtained from the character table, then, all that is needed is a knowledge of the division rings that occur. The main result of the paper identifies the division rings in the simple component corresponding to the irreducible characters of G.
TL;DR: In this article, it was shown that each matrix over a principal ideal ring is equivalent to some diagonal matrix, and partial results on the uniqueness of the diagonal form were obtained by specializing some general properties about simultaneous decompositions of a projective module and a homomorphic image.
TL;DR: In this article, it was shown that if R is a regular right noetherian ring and R{X} is the free associative algebra on the set X, then Kn(R) = Kn{X}, where Kn refers to the Quillen K-theory.
Abstract: The object of this article is to establish the following result (Corollary 3.9 below): If R is a regular right noetherian ring and R{X} is the free associative algebra on the set X, then Kn(R) = Kn(R{X}), where Kn refers to the Quillen K-theory. The result can be stated in the equivalent form that Hn(G1(R),Z) = Hn(G1(R{X}),Z). From this result it follows that if F is a free ring without unit, then Kn(F) = 0, whence free rings are acyclic models for Quillen K-theory (3.11 below). This result in turn enables us to complete Anderson's work [1] in identifying the Quillen K-theory [11] and the K-theory proposed by Gersten [7] and Swan [18] for all rings. We also establish that the natural transformation Kn(R) → Kn k-v(R) between the Quillen theory and the K-theory of Karoubi and Villamayor is an isomorphism if R is a supercoherent (Definition 1.2) and regular (Definition 1.3) ring. From this result we can gain some information about the K-theory of group rings of free products of groups (Theorem 5.1).
TL;DR: In this article it was shown that a large class of algebras have regular central polynomials and that these are called regular central regular regular regular ideals of rings.
TL;DR: An algorithm is presented which allows simple determination of a ring sum realization using logic array notation, and which can be used to find minimum cost polarities and a second algorithm which allows nonexhaustive and near-optimal handling of functions with DON'T CARE conditions.
Abstract: Reordering the terms of a Reed-Muller or ring sum expansion of a switching function expressed in terms of the Boolean ring operations AND and EXCLUSIVE OR in a more natural way exploits the similarities between these expressions and unate functions and displays mathematical structure which apparently has not been noted before. McNaughton's n orderings on the n cube appear in a new setting and lead quickly to simple but geometrically satisfying theorems dealing with a matrix of coefficients of all 2n ring sum expressions for various polarities of inputs. The structure of these matrices for minterms, implicants, and functions are shown to have simple and attractive forms. An algorithm is presented which allows simple determination of a ring sum realization using logic array notation, and which can also be used to find minimum cost polarities. A second algorithm allows nonexhaustive and near-optimal handling of functions with DON'T CARE conditions.
TL;DR: In this paper, the authors studied families of unipotent algebraic groups over integral rings and proved that the space of such families is isomorphic to an affine space over the base.
Abstract: In this paper we study families of unipotent algebraic groups over integral rings. The main results relate to the geometry of such families. In particular, we prove that, under some hypotheses, the space of such a family is isomorphic to an affine space over the base. We give counterexamples showing that in the case of an arbitrary base ring the basic facts of the theory of unipotent algebraic groups over a field cease to be true. For a certain class of the group schemes that we consider we prove results on cohomology, extensions and deformations.
01 Jan 1974
TL;DR: In this paper, it was shown that for each integer k^2, there is a non-Noetherian unique factorization domain of arbitrary characteristic, which is the same dimension as the polynomial ring A(x"^}) and the abelian group of torsion-free rank a.
Abstract: Let R be a commutative ring with identity and let G be an abelian group of torsion-free rank a. If {Xx} is a set of indeterminates over R of cardinality a, then the group ring of G over R and the polynomial ring A({x"^}) have the same (Krull) dimension. The preceding result and a theorem due to T. Parker and the author imply that for each integer k^2, there is a »c-dimensional non-Noetherian unique factorization domain of arbitrary characteristic.
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TL;DR: In this article, it was shown that rings are associative and have a unit unit and subrings are assumed to have the same unit, and ring homomorphisms are unitary, as are modules.
Abstract: §1 Statement of Results Rings are associative and have a unit and subrings are assumed to have the same unit. Ring homomorphisms are unitary, as are modules. Ideal means two-sided ideal.
TL;DR: In this article, a mathematical analysis is made of the optimal siting of concentric ring roads in a city so as to minimize the average distance travelled off the rings, and it is found that although such ring roads are more effective if they are characterized by a higher speed of travel than the rest of the city, their optimal location is largely independent of such speed.
Abstract: A mathematical analysis is made of the optimal siting of concentric ring roads in a city so as to minimize the average distance travelled off the rings. It is found that although such ring roads are more effective if they are characterized by a higher speed of travel than the rest of the city, their optimal location is largely independent of such speed. Most of the benefit of using a series of concentric ring roads can be obtained from one or two well-placed ring roads. Optimization with respect to the average overall travel distance is also considered.
TL;DR: In this article, the equivalence of Hori and Deprit's Lie Transform algorithm for non-canonical as well as canonical transformations has been established and a formula relating directly the two generating functions (or vector fields) is presented.
Abstract: The Lie transform method used in Perturbation Theory is based upon an intrinsic algorithm for transforming functions or vector fields by a transformation close to the identity. It can thus be viewed as a specialization of methods and results of differential geometry as is shown in the first part of this paper. In a second part we answer some of the questions left open in connection with the equivalence of the algorithms proposed by Hori and Deprit. From a formal point of view, the methods are shown to be equivalent for non-canonical as well as canonical transformations and a formula relating directly the two generating functions (or vector fields) is presented (formula (5.17)). On the other hand, the equivalence is shown to hold also in the ring ofp-differentiable functions.
TL;DR: In this article, the impulse response of an annular piston mounted in an infinite planar rigid baffle was evaluated using the double surface integral representation of the radiation impedance, and the results were shown to be in agreement with those of earlier investigators.
Abstract: An approach is presented to evaluate the impulse response of an annular piston, which is mounted in an infinite planar rigid baffle. Since the impulse response is defined to be the time‐dependent force on the ring, which results from an impulsive velocity of the ring, the impulse response and radiation impedance of the ring are a Fourier transform pair. A closed‐form expression for the impulse response of a ring of any size is developed from the well‐known double surface integral representation of the radiation impedance. Various integral and asymptotic expressions for the radiation impedance of the ring are derived from the impulse response, and the results are shown to be in agreement with those of earlier investigators. Several numerical results are also presented to show the effect of changing the ring size on the impulse response, and on the radiation resistance and reactance of the ring.
TL;DR: Theorem 3.1 as discussed by the authors states that the additive l-group of an f-ring determines the ring structure, and it is shown that each f-multiplication of S is determined by a homomorphism of S + into (S)+.
Abstract: The intent of this paper is to show that the additive l-group of an f-ring S determines the ring structure. This is why there are so many papers that simply extend known results for abelian l-groups to f-rings. Theorem 3.1 asserts that there is a one-to-one correspondence between the f-multiplications on S and a set of homomorphisms from the positive cone of the l-group S into the positive cone of the ring (S) of polar preserving endomorphisms of the l-group S. In fact, each f-multiplication of S is determined by a homomorphism of S + into (S)+.
01 Feb 1974
TL;DR: In this paper, it was shown that if the ring of polynomials over a commutative ring R is semi-hereditary then R is von Neumann regular.
Abstract: It is shown that if the ring of polynomials over a commutative ring R is semihereditary then R is von Neumann regular. This is the converse of a theorem of P. J. McCarthy. P. J. McCarthy [2] has recently proved that for the ring of polynomials R[x] over a commutative ring to be semihereditary, it is sufficient that R be von Neumann regular. The purpose of this note is to show that this condition actually characterizes von Neumann regular rings. The lattice of ideals of a commutative von Neumann regular ring is distributive, but not all such rings are von Neumann regular. If the lattice of ideals of R[x] is distributive then the lattice of ideals of R is, since R is a homomorphic image of R[x], In the process of proving the converse of McCarthy's theorem, we show that this latter condition characterizes von Neumann regular rings. Summarizing: Theorem. The following are equivalent for a commutative ring R: 1. R is von Neumann regular. 2. R[x] is semihereditary. 3. R[x] has a distributive lattice of ideals. Proof. 1 implies 2 is McCarthy's theorem. The fact that a commutative semihereditary ring has a distributive lattice of ideals may be found in [l], which yields 2 implies 3. To show 3 implies 1, we use the fact that a ring R has a distributive lattice of ideals if and only if, for r, s £ Rt (r :s) + (s:r)= R where (s :r) = ¡x £ R | sx £ rR\ [l]. The above statement is easily seen to be equivalent to the existence of u, v, and w € R with: r(l u) = sv and su = rw. Now, let a £ R. We must show a R = aR. The fact that R[x] has a distributive lattice of ideals yields u(x), v(x) and w(x) with: Received by the editors October 9, 1973. AMS (MOS) subject classifications (1970). Primary 16A30, 13F20; Secondary 13A15.