Victor Camillo

Armendariz rings and gaussian rings

We prove a number of results concerning Armendariz rings and Gaussian rings. Recall that a (commutative) ring R is (Gaussian) Armendariz if for two polynomials f,g∈R[X] (the ideal of R generated by the coefficients of f g is the product of the ideals generated by the coefficients of f and g) fg = 0 implies a i b j=0 for each coefficient a i of f and b j of g. A number of examples of Armendariz rings are given. We show that R Armendariz implies that R[X] is Armendariz and that for R von Neumann regularR is Armendariz if and only if R is reduced. We show that R is Gaussian if and only if each homomorphic image of R is Armendariz. Characterizations of when R[X] and R[X] are Gaussian are given.

Exchange rings, units and idempotents

An associative ring R with identity is semiperfect if and only if every element of R is a sum of a unit and an idempotent, and R contains no infinite set of orthogonal idempotents. A ring which contains no infinite set of orthogonal idempotents is an exchange ring if and only if every element is a sum of a unit and an idempo-tent

COMMUTATIVE RINGS WHOSE ELEMENTS ARE A SUM OF A UNIT AND IDEMPOTENT

ABSTRACT As defined by Nicholson a (noncommutative) ring is a clean ring if every element of is a sum of a unit and an idempotent. Let be a commutative ring with identity. We define to be a uniquely clean ring if every element of can be written uniquely as the sum of a unit and an idempotent. Examples of clean rings (uniquely clean rings) include von Neumann regular rings (Boolean rings) and quasilocal rings (with residue field ). A ring is a clean ring or uniquely clean ring if and only if is. So every zero-dimensional ring is a clean ring, but a zero-dimensional ring is a uniquely clean ring if and only if is a Boolean ring.

A CHARACTERIZATION OF UNIT REGULAR RINGS

It is proved that a ring R is unit regular if and only if every element a of R can be written as e + u such that aR ∩ eR = 0, where e is an idempotent and u a unit in R.

Semigroups and rings whose zero products commute

Let S be a semigroup with zero 0 and let n ≥ 2. We say that S satisfies for each permutation σ ∈ S n A ring R satisfies ZCn if (.R, .) satisfies ZCn. We show that if S satisfies ZCn for a fixed n ≥ 3, then S also satisfies ZCn+1, but we give an example of a ring R with identity which satisfies ZC2 but does not satisfy ZC3 We show that a semigroup with no nonzero nilpotents satisfiesZCn for all n ≥ 2 and investigate rings that satisfy ZCn.

Stable range one for rings with many idempotents

An associative ring R is said to have stable range 1 if for any a, b e R satisfying aR + bR = R, there exists y e R such that a + by is a unit. The purpose of this note is to prove the following facts. Theorem 3: An exchange ring R has stable range 1 if and only if every regular element of R is unit-regular. Theorem 5: If R is a strongly 7r-regular ring with the property that all powers of every regular element are regular, then R has stable range 1. The latter generalizes a recent result of Goodearl and Menal [5]. Let R be an associative ring with identity. R is said to have stable range 1 if for any a, b E R satisfying aR + bR = R, there exists y E R such that a + by is a unit. This definition is left-right symmetric by Vaserstein [9, Theorem 2]. Furthermore, by a theorem of Kaplansky, all one-sided units are two-sided in rings having stable range 1 (cf. Vaserstein [10, Theorem 2.6]). It is well known that a (von Neumann) regular ring R has stable range 1 if and only if R is unit-regular (see, for example, Goodearl [4, Proposition 4.12]). Call a ring R strongly 7-regular if for every element a E R there exist a number n (depending on a) and an element x E R such that an = an+lx. This is in fact a two-sided condition [3]. It is an open question whether all strongly 7r-regular rings have stable range 1. Goodearl and Menal [5] proved that strongly 7r-regular rings are unit-regular and, hence, have stable range 1 (Theorem 5.8, p. 278). In this note we first extend the above result for von Neumann regular rings to a larger class of rings, which includes all strongly 7r-regular rings, 7r-regular rings, von Neumann regular rings, and algebraic algebras. As an application of this, we prove that a strongly 7r-regular ring R has stable range 1 if powers of every regular element are regular. The latter is a generalization of the abovementioned result of Goodearl and Menal for strongly 7r-regular regular rings. As one can see from our proofs, rings in these classes have a large supply of idempotents. Throughout, R stands for an associative ring with identity and J(R) for the Jacobson radical of R. Modules are unitary right R-modules except otherwise specified. For other undefined terms, readers are referred to [4]. Let MR be a right R-module. Following Crawley and Jonsson [2], MR is said to have the exchange property if for every module AR and any two Received by the editors January 24, 1994 and, in revised form, May 3, 1994; originally communicated to the Proceedings of the AMS by K. A. Goodearl. 1991 Mathematics Subject Classification. Primary 1 6D70, 1 6P70.

Coherence for polynomial rings

Let II= IIR, be an arbitrary product of copies of R,. We say R is right n-coherent if every finitely generated submodule of Z7 is finitely presented. This notion is called strong coherence in [7], where it seems to have been invented. Here we obtain a new characterization of n-coherence, and study Z7coherence for polynomial rings. Using Goldie’s Theorem, we show that if R is a semiprime two-sided noetherian ring then R[S] is right n-coherent for any set of variables S (Theorem 6). We also obtain the same result for two-sided noetherian rings R, not assumed to be semiprime, provided R contains an uncountable field. Call ring R a left *-ring (star ring) if it has the property that HomA , RR) = ( )* takes finitely generated left R-modules to finitely generated right R modules. The above ideas are connected by the following:

Balanced rings and a problem of Thrall

Balanced ring is defined and related to Thralls QF-1 rings. Several theorems are obtained which show that balanced rings enjoy strong homological and chain conditions. The structure of commutative balanced rings is determined. Also, the structure of commutative artinian QF-1 rings is gotten. This is a generalization of a theorem of Floyd. Introduction. If M is a right R-module, then M is a natural left module over its endomorphism ring S. We call T= Ends M the BiEndomorphism ring of M, and the elements of Tare called BiEndomorphisms (notation BiEnd M). The mapping: -q: R -> BiEnd MR, ?: a-> ad, where (x)ad = xa, Vx E M is a ring homomorphism and ker 71 is the annihilator of M (notation ker Xj = annR M). The elements of BiEnd MR of the form ad are called right multiplications of M. Every element of BiEnd MR is a right multiplication iff the natural map D: R -? BiEnd M is surjective, that is, a ring epimorphism. In this case, following Faith [2], we say M is balanced. If M is balanced, we have a complete description of BiEnd MR, namely BiEnd MRR/annR M. Naturally, this is not always the case, as is well known. In this paper we study rings for which every right R-module is balanced, and call R right balanced in this case. It appears that balanced tings have not been studied in this generality. Thrall [13] proposed the classification of finite dimensional algebras, called QF-1 algebras, having the property that every finitely generated faithful right R-module is balanced. The problem remains unsolved at the present, but there are results in special cases (Floyd [5], Fuller [7] and Morita [9]). The point of departure of this paper, and the idea which led to our main results is the observation that first, the QF-1 hypothesis, when assumed for general rings and their quotients, actually implies chain conditions, and second that the determination of BiEnd MR is what we want, for a general R-module M, not merely Received by the editors September 8, 1969. AMS Subject Classifications. Primary 1340, 1350.

QUASI-MORPHIC RINGS

A ring R is called left morphic if R/Ra ≅ l(a) for each a ∈ R, equivalently if there exists b ∈ R such that Ra = l(b) and l(a) = Rb. In this paper, we ask only that b and c exist such that Ra = l(b) and l(a) = Rc, and call R left quasi-morphic if this happens for every element a of R. This class of rings contains the regular rings and the left morphic rings, and it is shown that finite intersections of principal left ideals in such a ring are again principal. It is further proved that if R is quasi-morphic (left and right), then R is a Bezout ring and has the ACC on principal left ideals if and only if it is an artinian principal ideal ring.

NILPOTENT IDEALS IN POLYNOMIAL AND POWER SERIES RINGS

Given a ring R and polynomials f(x), g(x) ∈ R[x] satisfying f(x)Rg(x) = 0, we prove that the ideal generated by products of the coefficients of f(x) and g(x) is nilpotent. This result is generalized, and many well known facts, along with new ones, concerning nilpotent polynomials and power series are obtained. We also classify which of the standard nilpotence properties on ideals pass to polynomial rings or from ideals in polynomial rings to ideals of coefficients in base rings. In particular, we prove that if I ≤ R[x] is a left T-nilpotent ideal, then the ideal formed by the coefficients of polynomials in I is also left T-nilpotent.