Showing papers on "Ring (mathematics) published in 1977"

Lifting idempotents and exchange rings

Abstract: Idempotents can be lifted modulo a one-sided ideal L of a ring R if, given x e R with x-x2 cL, there exists an idempotent e c R such that e x E L. Rings in which idempotents can be lifted modulo every left (equivalently right) ideal are studied and are shown to coincide with the exchange rings of Warfield. Some results of Warfield are deduced and it is shown that a projective module P has the finite exchange property if and only if, whenever P = N + M where N and M are submodules, there is a decomposition P = A @ B with A S N and B C M. In 1972 Warfield showed that if M is a module over an associative ring R then M has the finite exchange property if and only if end M has the exchange property as a module over itself. He called these latter rings exchange rings and showed (using a deep theorem of Crawley and Jonsson) that every projective module over an exchange ring is a direct sum of cyclic submodules. Let J(R) denote the Jacobson radical of R. Warfield showed that, if R/J(R) is (von Neumann) regular and idempotents can be lifted modulo J(R), then R is an exchange ring and so generalized theorems of Kaplansky and Muller. The main purpose of this paper is to prove the following theorem: A ring R is an exchange ring if and only if idempotents can be lifted modulo every left (respectively right) ideal. The properties of these rings are examined in the first section and the theorem is proved in the second section. The theorems of Warfield are then easily deduced and a new condition that a projective module have the finite exchange property is given. 1. Suitable rings. In this section, the rings of interest are defined, some of their properties are deduced, and several examples are given. All rings are assumed to be associative with identity and J(R) denotes the Jacobson radical of a ring R. 1.1. PROPOSITION. If R is a ring, the following conditions are equivalent for an element x of R. Received by the editors December 2, 1975. AMS (MOS) subject classifications (1970). Primary 16A32, 16A64; Secondary 16A30, 16A50.

E∞ ring spaces and E∞ ring spectra

Abstract: ? functors.- Coordinate-free spectra.- Orientation theory.- E? ring spectra.- On kO-oriented bundle theories.- E? ring spaces and bipermutative categories.- The recognition principle for E? ring spaces.- Algebraic and topological K-theory.- Pairings in infinite loop space theory.

Group inverses and Drazin inverses of bidiagonal and triangular Toeqlitz matrices

Abstract: Necessary and sufficient conditions are given for the existence of the group and Drazin inverses of bidiagonal and triangular Toeplitz matrices over an arbitrary ring.

Flat ideals II

Abstract: This paper is concerned with the relationships that hold between finiteness and divisibility properties of flat ideals of integral domains Brought out often is the role of the arithmetic of a ring in the finiteness of its flat ideals

A note on the group structure of unit regular ring elements

Abstract: 1* Introduction* It is well-known that [15, 7] a ring R is strongly regular if and only if every aeR is a group member. In this note we shall use the basic theorem for group members in a ring to show locally that a ring element a e R (with unity) is unit regular exactly when there is a unit ue R and a group G in R such that a e uG. Hence unit regular rings are, as it were locally a "rotated" version of strongly regular rings. We remind the reader that a ring R is called regular if for every aeR, aeaRa; strongly regular if for every aeR, aeaR, and unit regular if for every aeR, there is a unit u e R such that ana = a [3]. Similar definitions hold locally. A ring with unity is called finite if ah = 1 implies ha — 1. Any solution a~ to axa = a is called an inner or 1-inverse of [1], while any solution a to axa = a and xax = x is called a reflexive or 1-2 inverse of a. For idempotents e and / in R, e ~ / denotes the equivalence in

A constructive approach to commutative ring theory

Abstract: The development of a system in MACSYMA for solving ring problems is described. Fundamental algorithms are given for expressing ideals in canonical form, and concrete examples are demonstrated.

Unsolvability of the endomorphic reducibility problem in free nilpotent groups and in free rings

Abstract: UNSOLVABILITY OF THE ENDOMORPHIC REDUCIBILITY PROBLEM IN FREE NILPOTENT GROUPS AND IN FREE RINGS V. A. Roman'kov UDC 519.4 We say that the endomorphic reducibility problem is solvable in a group or ring ~ if there exists an algorithm which solves the occurrence problem for any element ~G in the set of its endomorphic images. In discussing this problem in [i], R. Lyndon noted that it stems from A. Tarski's well-known problem concerning the decidability of elementary theories of free non-Abelian groups. It has been shown that there exist finitely presented groups with an unsolvable endomorphic reducibility problem (see, e.g., [2]). The known partial results on this problem for free groups are affirmative. Suppose Q is a free group (free ring) of countable rank in some variety. Then the endomorphic reducibility problem for ~ is equivalent to that of the solvability in ~ of equations of the form

Digital filtering using polynomial transforms

Abstract: We define discrete transforms in a ring of polynomials. These polynomial transforms have the circular convolution property and can be used for the fast computation of 2-dimensional cyclic convolutions. This yields efficient algorithms for the implementation of 1- and 2-dimensional digital filters.

Semimaximal rings of finite type

Abstract: Necessary and sufficient conditions are given for a semimaximal ring to have finitely many indecomposable nonisomorphic finitely generated modules that are torsion-free in the sense of Bass.Bibliography: 13 titles.

The generalized Kronecker function ring and the ring R(X).

Abstract: This paper extends the concept of Kronecker function rings from integral domains to rings with zero divisors. To accomplish this we use Marot's property (P) and a new property which we label äs (A). The first part of the article is devoted to developing results on properties (P) and (A), and showing that the class of rings satisfying these properties is quite large. The main theorem, a generalization of a result about integral domains due to J. Arnold, is given in §3. We prove that if R is a ring having properties (P) and (A), and if R is the Kronecker function ring of R, then the following are equivalent: (1) R is a Prüfer ring; (2) R(X) = R; (3) R(X) is a Prüfer ring; and (4) R is a regulär quotient ring of R[X]. The class of rings for which this result holds is large. For example, it properly contains the set of commutative semihereditary rings. In the final section we construct an example, using the idealization of an -module, to show that the main theorem does not hold if property (A) is deleted from the hypothesis.

On the Kerr–Tomimatsu–Sato family of spinning mass solutions

Abstract: Some analytic properties of the Kerr–Tomimatsu–Sato family of solutions with arbitrary integer deformation parameter δ for gravitational fields of spinning masses are studied. It is shown that all ring singularities are of first order and all ergosurfaces are simple zeros of metric functions A.

Noetherian fixed rings.

Abstract: by a group of automorphisms of R. This paper explores what happens when the group is finite and the fixed ring S is assumed to be Noetherian Easy examples show that R may not be Noetherian; however, in this paper it is shown that R is Noetherian with some rather natural assuptions. More precisely we prove the Theorem 2: Let S be a semiprime ring. Assume that G is a finite group of automorphisms of 5 and that S has no | G [-torsion. If S° is left noetherian then S is left noetherian.

A Ring of Delay Operators with Applications to Delay-Differential Systems

Abstract: A ring of delay operators is used to obtain a representation of the solution of systems of linear delay-differential equations. With the aid of this representation an algebraic rank-test is obtained for the $R^n $-controllability of the systems.

A countable self-injective ring is quasi-Frobenius

Abstract: A countable dimensional self-injective algebra is Artinian. There is an application to self-injective twisted group algebras. It has been known for some time that a countable self-injective ring is semilocal (see for example [8]). In this paper we show that such a ring is in fact quasi-Frobenius. Special cases of this result have been proved previously, for example if the ring is also regular [3] or if it is a group algebra [8]. My thanks to Ken Louden for his help in the preparation of this paper. Unless stated otherwise, all rings are associative with a unity. If S is a subset of a ring R, we denote its left annihilator in R by lR(S). Theorem 1 (Faith [1]). A ring is quasi-Frobenius if it is right self-injective and satisfies the descending chain condition on left annihilators. Proposition 2. Let R be a subring of S. Suppose that Ss is injective, RS is flat and SR is free. Then RR is injective. Proof. The proof is left to the reader. Theorem 3. Every countable subring of a quasi-Frobenius ring is contained in a countable quasi-Frobenius subring. Conversely, if every countable subring of a ring is contained in a quasi-Frobenius subring, then the ring is quasi-Frobenius. Proof. Suppose first that F is a quasi-Frobenius ring and A is a countable subring. We construct a sequence of subrings A = R0 e Rx E R2 C ■ • • C T inductively as follows. Given Rk, consider all «-tuples {a,, . . . , an) of elements of Rk as « ranges over the positive integers. If an E ax T + • • • + a„-\T choose xx, x2, . . . , xn_x E T so that an = axxx + ■ ■ ■ + an_xxn_x. If an G axT + • • • + an_xT, choose xn G T so that xna¡ = 0, ; = 1, 2, . . . , n — 1, and x„an =£ 0. Now do the same for the left ideal generated by ax, a2, . . . , an_x. Let Rk + X be the subring of T generated by Rk and all the x's obtained. Let R = U " i^,Clearly A E R and R is a countable subring, so Received by the editors June 23, 1976. AMS (A/OS) subject classifications (1970). Primary 16A52; Secondary 16A26.

Positive derivations on f-rings

Abstract: Throughout this paper A will denote an f-ring i.e. a lattice-ordered ring in the sense of Birkhoff and Pierce (1956) in which for all x, y, z ∈ A, x ∧ y = 0 implies x ∧ zy = 0 = x ∧ yz.

Zero divisors and idempotents in group rings

Abstract: 1. Introduction. Let kG denote the group algebra of a group G over a field k. In this paper we are first concerned with the zero divisor conjecture: that if G is torsion free, then kG is a domain. Recently, K. A. Brown made a remarkable breakthrough when he settled the conjecture for char k = 0 and G abelian by finite (2). In a beautiful paper written shortly afterwards, D. Farkas and R. Snider extended this result to give an affirmative answer to the conjecture for char k = 0 and G polycyclic by finite. Their methods, however, were less successful when k had prime characteristic. We use the techniques of (4) and (7) to prove the following:Theorem A. If G is a torsion free abelian by locally finite by supersoluble group and k is any field, then kG is a domain.

Related representation theorems for rings, semi-rings, near-rings and semi-near-rings by partial transformations and partial endomorphisms

Abstract: Fundamental statements for (associative) rings are that (a) the endomorphisms of each commutative group (U, +) form a ring and (b) eachring may be embedded in such a ring of endomorphisms. In order to generalise these theorems to groups and rings whose addition may not be commutative, one has to deal with partial endomorphisms. But thesering-theoretical Theorems 4a and 4b turn out to be specialisations of similarones for semi-near-rings, near-rings and semirings, developed here inSection 2 after some preliminaries on semi-near-rings in Section 1. A glance at Definition 1 and the ring-theoretical theorems and remarks at the end of Section 2 may give more orientation.

Prime differential ideals in differential rings

Abstract: Publisher Summary Throughout this paper, all rings are commutative with identity and all ring homomorphisms preserve the identity. Also, all differential rings are ordinary, i.e., possess a single derivation operator which is suppressed from the notation. If A is a differential ring and x ∈ A, then x(n) denotes the nth derivative of x; we note that x(0) = x. A subset of a differential ring is called differential if it contains the derivative of each of its elements.

On Weitzenböck’s theorem in positive characteristic

Abstract: Let k be an algebraically closed field and letf: Ga GL(V) be a finite-dimensional k-rational representation of the additive group Ga If the subspace of Ga-fixed points in V is a hyperplane, then the ring of Ga-invariant polynomial functions on V is finitely generated over k This result is an analog of a classical theorem of Weitzenbock, a modem proof of which has been given by C S Seshadr