Joe W. Fisher

Semiprime Skew Group Rings

In this paper we prove that if G is a finite group of automorphisms acting on a semiprime ring R such that R has no additive ] G j-torsion, then the skew group ring R*G is also semiprime. The result was heretofore known in such special cases as when G is finite abelian, R is Goldie, or R satisfies a polynomial identity [I]. Our technique of proof is to show first that (even in the presence of 1 G I- torsion or G possibly infinite) every nonzero ideal in R*G contains a nonzero element whose support consists of inner automorphisms, where inner is used in the sense of Kharchenko [5]. Using this we are able to show that R*G is semiprime provided that R is prime and B*Ginn is semiprime where Glnn is the subgroup of inner automorphisms and B is the algebra of the group in the quotient ring of R with respect to the filter of two-sided ideals. Next we show that indeed B*Ginn is semiprime when R has no additive 1 G j-torsion. Finally, in order to complete the proof, we make a reduction from R semiprime to R prime. Along the way we prove that if G (possibly infinite) is outer and

Rings generated by their units

In 1953 K. Wolfson [14] and in 1954 D. Zelinsky [15] showed, independently, that every element of the ring of all linear transformations of a vector space over a division ring of characteristic not 2 is a sum of two nonsingular ones. Motivated by this result, in 1958, Skornyakov [12, p. 1671 conjectured that all regular rings in which 2 is a unit are generated by their units. The conjecture was finally settled in the negative by G. Bergman [7] in 1974. At about the same time [6], we proved the conjecture in the affirmative for a large class of regular rings; that being, the class of regular rings which are strongly a-regular. Included in this class are those regular rings with primitive factor rings Artinian. In fact, we prove in Theorem 1 that they are unit regular from which the generation by units follows immediately. As a corollary we obtain that regular rings which satisfy a polynomial identity are unit regular. Our best theorem-Theorem 3 -states that if R is strongly n-regular and 2 is a unit in R, then each element of R can he expressed as a sum of two units. The class of stronlgy p-regular rings includes algebraic algebras, locally Artinian rings, and rings with prime factor rings Artinian. Throughout, R will denote a ring with unity element. A ring R is called (van Neumann) regular if for each a in R there exists an x in R such that a = axa. An element a in R is unit regular if there exists a unit u in R such that a = am and R is called unit regular if each element of R is unit regular. In [3] G. Ehrlich proved that the class of unit regular rings includes semisimple Artinian rings, regular rings with no nonzero nilpotent elements, and regular group algebras over fields. The following theorem produces a sufficient condition for regular rings to he unit regular.

Semiprime ideals in rings with finite group actions

LetR be a ring, G a finite group of automorphisms acting on R, and RG the-fixed subring of R. We prove that if R is semiprime with no additive ¦ G¦-torsion, then R is left Goldie if and only if RG is left Goldie. By coupling this with an examination of the prime ideal structures of RG and R, we are able to prove that if ¦G ¦ is invertible in R and RG is left Noetherian, then R satisfies the-ascending chain condition on semiprime ideals, every semiprime factor ring of R is left Goldie, and nil subrings of R are nilpotent. For the pair RG and R, we also consider various other properties of prime and maximal ideals such as lying over, going up, going down, and incomparability.

Invariants of finite cyclic groups acting on generic matrices

A classical theorem of E. Noether asserts that if R is a commutative ring, finitely generated over a field k, and G is any finite group of k- automorphisms of R, then the fixed ring (or ring of invariants) R” is also finitely generated. The question naturally arises as to what extent Noether’s theorem can be generalized to the noncommutative case. If R is also Noetherian and IG/ ~ ’ E k, all is well: RG is finitely generated, a result of Montgomery and Smail [6]. However, it is false in general, even for Pf rings [6]. Moreover, a recent result in Dicks and Formanek [l] (and, somewhat later, Kharchenko [4]), shows that almost the opposite of Noether’s theorem holds in the free aIgebra. That is, they prove that if G acts linearly on the free algebra F= k(xj,..., x,>, then P is finitely generated if and only if G acts by scalar matrices. In the present paper we consider the analogous problem for an algebra of generic matrices. That is, let U = k(X i,..., X,f be the generic matrix algebra generated over a field k by the m x m (m 2 2) generic matrices X1,..., X, (da 2). Let G act linearly on U; that is, for each gE G, Xp = c, cliiXi, for uij E k. Thus g corresponds to the dxd matrix A = (a,). If G consists of scalar matrices and \G/ -’ E k, then UC is always finitely generated. For, consider the free algebra F= k(x, ,..., xd> with the same action; since I/= ir’, a homomorphic image of F, it follows that v =

ON CENTER-LIKE ELEMENTS IN RINGS

In a paper with a similar title Herstein has considered the structure of in prime rings which contain an element a which satisfies either (a x 0 or is in the center of R for each x in R. We consider the structure of rings which contain an element a which satisfies powers of certain higher commutators. The two types which