Goldie Ring Research Articles

Marubayashi-Krull orders and strongly graded rings

1. INTRODUCTION The aim of this note is to give sufficient conditions for a strongly Z-graded ring to be a Marubayashi-Krull order (cf. Theorem 3.10). Concrete examples of strongly E-graded rings are skew group rings over Z and generalized Rees rings. We also prove that under the same conditions the positive part of a strongly Z-graded ring is a Marubayashi-Krull order (cf. Theorem 4.1). Finally, we examine when these conditions may be weakened. 2. PRELIMINARIES A ring R is said to be a strongly B-graded ring if there exists a family of additive subgroups R, (n E Z) of R such that R = OnpLRn and wm =Rn+m for all n, m E Z. The elements of h(R) = lJneh R, are called the homogeneous elements of R. If x E R,, then x is said to be homogeneous of degree n. If x = C x, (x, E R,), then the elements x, are called the homogeneous components of x. Let I be an ideal of R. Denote by Ig the ideal generated by the homogeneous elements contained in I. I is said to be graded if I = Ig. In this case Z= RI, =Z,R, where I, = Zn R,. Details about strongly graded rings may be found in [lo]. If I is an ideal of R, then C(I) denotes the set of elements regular modulo I. If C(I) is a left and right Ore set in R, then Rcc,, denotes the left and right localization of R towards this Ore set C(1). If R is a prime Goldie ring, then Q(R), the classical ring of quotients of R, is a simple Artinian ring. The Asano overring of R is defined as S(R) = {x E Q(R) 1 Ix c R for some nonzero ideal I of R}. A prime Goldie ring R is said to be a Marubayashi-Krull order (or for short an M-Krull order) if:

Sums of two regular elements

A well-known fact concerning a prime right Goldie ring R, proved in [4, Section 5], is that an essential right ideal is generated by the regular elements which it contains. There is a modification of that proof which shows that each element of R is the sum of at most two regular elements. This suggested that the recent results of Chatters and Ginn [1] concerning rings generated by their regular elements might possibly be refined a little, since their arguments actually show that elements of R are sums of at most three regular elements.

Rings generated by their regular elements

The units of a ring R are defined by means of a multiplicative property, but in many cases they generate R additively. For example, it is shown in [5, Proposition 6] that if R is a semi-simple Artinian ring then every element of R is a sum of units if and only if the ring S = ℤ/2ℤ⊕ℤ/2ℤ is not a direct summand of R, where ℤ denotes the ring of integers. The theme of this paper is to investigate the corresponding situation concerning regular elements, i. e. elements which are not zero-divisors. We show that if R is a semi-prime right Goldie ring then every element of R is a sum of regular elements if and only if R does not have the ring S defined above as a direct summand (Corollary 2.9). We also characterise those Noetherian rings R such that every element of R is a sum of regular elements (Theorem 2. 6). The characterisation is in terms of the nature of certain prime factor rings of R, and it is again the presence of the ring S, this time in a particular way as a factor ring of R, which prevents R from being generated by its regular elements. If R has no non-zero Artinian one-sided ideals or if 2 is a regular element of R, then every element of R is a sum of regular elements (Corollaries 2. 5 and 2. 7). As an application we show in Section 3 that, for many Noetherian rings R, the set of elements of R which are divisible by every regular element of R is a two-sided ideal of R.

On quasiprojective covers

The main results determine the Goldie dimension of superfluous submodules of semiprime left Goldie rings and apply this to the study of quasiprojective covers of torsion free modules. Conditions are given to guarantee that a quasiprojective cover of a torsion free module is an isomorphism. A class of nonperfect rings is given such that finitely generated singular modules possess quasiprojective covers.

INTEGRALLY CLOSED RINGS

This paper studies integrally closed rings. It is shown that a semiprime integrally closed Goldie ring is the direct product of a semisimple artinian ring and a finite number of integrally closed invariant domains that are classically integrally closed in their (division) rings of fractions. It is shown also that an integrally closed ring has a classical ring of fractions and is classically integrally closed in it. Next, integrally closed noetherian rings are considered. It is shown that an integrally closed noetherian ring all of whose nonzero prime ideals are maximal is either a quasi-Frobenius ring or a hereditary invariant domain. Finally, those noetherian rings all of whose factor rings are invariant are described, and the connection between integrally closed rings and distributive rings is examined. Bibliography: 13 titles.

Quotient rings of semiprime rings with bounded index

We say that a ring R has bounded index if there is a positive integer n such that an = 0 for each nilpotent element a of R. If n is the least such integer we say R has index n. For example, any semiprime right Goldie ring has bounded index, and so does any semiprime ring satisfying a polynomial identity [10, Theorem 10.8.2]. This paper is mainly concerned with the maximal (right) quotient ring Q of a semiprime ring R with bounded index. Several special cases of this situation have already received attention in the literature. If R satisfies a polynomial identity [1], or if every nonzero right ideal of R contains a nonzero idempotent [18] then it is known that Q is a finite direct product of matrix rings over strongly regular self-injective rings, the size of the matrices being bounded by the index of R. On the other hand if R is reduced (that is, has index 1) then Q is a direct product of a strongly regular self-injective ring and a biregular right self-injective ring of type III ([2] and [15]; the terminology is explained in [6]). We prove the following generalization of these results (see Theorems 9 and 11).

Two Results on Fixed Rings

Let $R$ be a semiprime ring, $G$ a finite group of automorphisms of $R$ and $B$ the algebra of the group. (A) If $R$ is left primitive and $B$ is $G$-simple then the fixed subring ${R^G}$ is left primitive. (B) If $B$ is semiprime and ${R^G}$ is a left Goldie ring, then $R$ can be embedded in a free left ${R^G}$-module of finite rank. Consequently if ${R^G}$ is left Noetherian, $R$ is left Noetherian.

Two results on fixed rings

Let R R be a semiprime ring, G G a finite group of automorphisms of R R and B B the algebra of the group. (A) If R R is left primitive and B B is G G -simple then the fixed subring R G {R^G} is left primitive. (B) If B B is semiprime and R G {R^G} is a left Goldie ring, then R R can be embedded in a free left R G {R^G} -module of finite rank. Consequently if R G {R^G} is left Noetherian, R R is left Noetherian.

Goldie conditions in finite normalizing extensions

Let S be a finite normalizing extension of a ring R. If M is an S module, then M has finite uniform dimension if and only if it has finite uniform dimension when considered as an R module. Consequently, when S is a right Goldie ring, R is also a right Goldie ring. Conversely, if R is a semiprime right Goldie ring and S is a prime ring, then S is a Goldie ring. Finally, when both S and R are semiprime right Goldie rings, the quotient ring of R embeds in the quotient ring of S.

An example of a Goldie ring whose matrix ring is not Goldie

An example of a Goldie ring whose matrix ring is not Goldie

On the structure of certain PP-rings

All rings considered are associative and have identity elements. By "Artinian rings", "Noetherian rings", "Goldie rings" and "hereditary rings" we shall mean rings which have the respective conditions on both sides. A ring R is a right PP-ring provided every principal right ideal is projective. Small [11], Theorem l, has proved that a right PP-ring which does not certain an infinite collection of orthogonal idempotents is also a left PP-ring. By a PPring we shall mean a ring which is both a right PP-ring and a left PP-ring. We shall call a PP-ring limited if it does not contain an infinite collection of orthogonal idempotents. Chatters [1], Theorem 3.1, has proved that any Noetherian PP-ring can be written as a finite direct sum of prime PP-rings and Artinian PP-rings. In an earlier theorem Levy [10], Theorem 4.3, had proved that any semiprime right Goldie PP-ring is a direct sum of prime rings. The primary purpose of this note is to investigate this theorem of Chatters and to extend it. If I is an ideal of a ring R then by 'CgR(I ) (respectively cg~(I)) we shall mean the set of elements c of R such that whenever reR and rc~I (creI) it follows that r~I. If S is a nonempty subset of R then l(S) and r(S) will denote the left and right annihilators of S, respectively. Our main theorem states that if R is a right Noetherian PP-ring with prime radical N then R is the direct sum of a semiprime ring and a ring with essential right socle if and only if '~(O)c_'Cg(K), where K =Nnl(N) (Theorem 3.7). A rather similar argument shows that if R is a PP-ring with prime radical N such that R/N is a Goldie ring and N is finitely generated both as a right ideal and as a left ideal then R is the direct sum of a semiprime ring and an Artinian ring (Theorem 4.3). Fuelberth and Kusmanovich [3], Theorem 3.12, have proved that a hereditary ring which has a semiprimary right and left quotient ring can be written as a finite direct sum of semiprimary rings and prime rings. Recall the Chatters in [1], Theorem4.1, has shown that a PP-ring R with prime r ad i ca lN has a semiprimary right quotient ring if and only if R/N is a right Goldie ring and

Baer Endomorphism Rings and Closure Operators

A Baer ring is a ring in which every right (and left) annihilator ideal is generated by an idempotent. Generalizing quite naturally from the fact that the endomorphism ring of a vector space is a Baer ring, Wolfson [5; 6] investigated questions such as when the endomorphism ring of a free module is a Baer ring, and when the ring of continuous linear transformations on a pair of dual vector spaces is a Baer ring. A further generalization was made in [7], where the question of when the endomorphism ring of a torsion-free module over a semiprime left Goldie ring is a Baer ring was treated.

Monomial conditions on rings

Drazin introduced the notion ofpivotal monomial, a condition on the evaluations of monomials in a ring, and characterized simple artinian rings as those primitive rings which have pivotal monomials. In this paper we consider monomial conditions related to pivotal monomials. The two major results are a characterization of prime Goldie rings in terms of pivotal monomials, and a characterization of the socle of a primitive ring in terms of generalized pivotal monomials.

On Maximal Torsion Radicals. III

Maximal torsion radicals of a strongly semiprime ring correspond to minimal prime ideals and can be used to characterize both strongly semiprime and semiprime Goldie rings.

Twisted group rings which are semi-prime Goldie rings

In this paper we examine when a twisted group ring,Rγ(G), has a semi-simple, artinian quotient ring. In §1 we assemble results and definitions concerning quotient rings, Ore sets and Goldie rings and then, in §2, we defineRγ(G). We prove a useful theorem for constructing a twisted group ring of a factor group and establish an analogue of a theorem of Passman. Twisted polynomial rings are discussed in §3 and I am indebted to the referee for informing me of the existence of [4]. These are used as a tool in proving results in §4.

On maximal torsion radicals. III

Maximal torsion radicals of a strongly semiprime ring correspond to minimal prime ideals and can be used to characterize both strongly semiprime and semiprime Goldie rings.

Strongly prime rings

A ring R is (right) strongly prime (SP) if every nonzero twosided ideal contains a finite set whose right annihilator is zero. Examples are domains, prime Goldie rings and simple rings; however, this notion is asymmetric and a right but not left SP ring is exhibited. All SP rings are prime, and every prime ring may be embedded in an SP ring. SP rings are nonsingular, and a regular SP ring is simple; since faithful rings of quotients of SP rings are SP, the complete ring of quotients of an SP ring is simple. All SP rings are coefficient rings for some primitive group ring (a generalization of a result proved for domains by Formanek), and this was the initial motivation for their study. If the group ring RG is SP, then R is SP and G contains no nontrivial locally finite normal subgroups. Coincidentally, SP rings coincide with the ATF rings of Rubin, and so every SP ring has a unique maximal proper torsion theory, and (0) and R are the only torsion ideals.( 1 ^{1} ) A list of questions is appended.

Regular Elements in Rings with Involution

The purpose of this paper is to determine when a symmetric element, regular with respect to other symmetries, is regular in the ring. This result is true for simple rings, for prime rings with either Goldie chain condition, and for semiprime Goldie rings. Examples are given to show that these results are the best that can be hoped for.

Goldie-Like Conditions on Jordan Matrix Rings

In this paper Goldie-like conditions are put on a Jordan matrix ring $J = H({R_n},{\gamma _a})$ which are necessary and sufficient for R to be a $\ast$-prime Goldie ring or a Cayley-Dickson ring. Existing theory is then used to obtain a Jordan ring of quotients for J.

Regular elements in rings with involution

The purpose of this paper is to determine when a symmetric element, regular with respect to other symmetrics, is regular in the ring. This result is true for simple rings, for prime rings with either Goldie chain condition, and for semiprime Goldie rings. Examples are given to show that these results are the best that can be hoped for. The structure of a ring with involution satisfying the property that no two nonzero symmetric elements annihilate one another was determined for 2-torsion. free rings by Lanski in [4], and in general, by Herstein in [2]. It happens, at least for semiprime rings which are 2-torsion-free, that this assumption on the symmetric elements implies that each nonzero symmetric element is regular in the whole ring. It is our purpose here to determine conditions sufficient to imply that any single symmetric element which annihilates no nonzero symmetric element is regular in the whole ring. Throughout this work, R will denote a ring with involution * and S = r e RIr*= rj, the set of symmetric elements of R. To avoid confusion, we may write S(A) for the symmetric elements of some ring A, with involution. We shall say that an element y is regular in S to mean that yt = 0 implies t = 0, for t E S. We are interested in determining when an element y e S, which is regular in S, must also be regular in R. Of course, we cannot hope to draw such a conclusion unless R is semiprime. A counterexample can be found easily by taking R @ N, for N a 2-torsion-free ring with trivial multiplication, and setting x* = -x for x e N. It is also easy to find examples where R has 1, and so N is not a direct summand. Let R = F[x, y. z]/(z2, zx, zy), for F a field with char F , 2, and set z*-z, x y and y*= x. One condition on R which allows us to conclude that elements regular in S are regular in R is given in our first observation. Received by the editors September 7, 1973. AMS(MOS) subject classifications (1970). Primary 16A28; Secondary 16A12, 16A46, 16A40, 16A38. (1) This work was supported by NSF Grant GP 38601. Copyright