A note on n-clean group rings

Associative rings with metabelian adjoint group

The following theorem is proved: Suppose R is an associative ring and suppose J is the Jacobson radical of R. Suppose that for all x1, * *, xn in R, there exists a word wx... (x1, * * x), depending on x1, * *, x,,, in which at least one xi (i varies) is missing, and such that xi ... x, = w ,... .n(x1, , x). Then J is a nil ring of bounded index and R/J is finite. It is further proved that a commutative nil semigroup satisfies the above identity if and only if it is nilpotent. In this paper, we investigate the structure of an associative ring R with the property that, for all xl, * * * , x, in R, xl ... x = wxi.... xn(xl, * * *, Xn), where w is a word, depending on xl, * * *, xn, and where some xi (i varies) is missing in w. For such a ring R, we prove the Jacobson radical J is a nil ring of bounded index. We also show that R/J is finite. Finally, we show that a commutative nil semigroup satisfies the above identity if and only if it is nilpotent. In preparation for the proofs of the main results, we first introduce DEFINITION 1. Let n be a positive integer, and let S be a semigroup. We say that S is a fln-semigroup if, for all xl, * * *, xn in S, there exists a word w .(x, *1 , xn) consisting of a product of the xi's with some x1 (j varies) missing, such that xl ... xn=wxl....xn((xl, * * *, xn). A ring R is called a fln-ring if its multiplicative semigroup is a fln-semigroup. LEMMA 1. Let S be a finite semigroup or a nilpotent semigroup. Then S is a fln-semigroup, for some n. PROOF. First, suppose S is finite, of order n, and suppose xl, Xn+1 are any elements of S. Then xi=xj for some i>j, and hence X1 ... Xn+1 = X1 ... Xi ... Xi-lXjXi+l Xn+1 = W(X1, . . , Xi-1 Xi+1 , Xn+1) Thus S is a fln+l-semigroup. Next, suppose S is nilpotent, say, Sm= (0). Presented to the Society, January 18, 1972; received by the editors August 10, 1971 and, in revised form, October 6, 1971. AMS (MOS) subject classifications (1970). Primary 16A38, 16A48; Secondary 20M10. The author was supported by a National Science Foundation Graduate Fellowship. @ American Mathematical Society 1973

We investigate properties of group gradings on matrix rings M n ( R ) , where R is an associative unital ring and n is a positive integer. More precisely, we introduce very good gradings and show that any very good grading on M n ( R ) is necessarily epsilon-strong. We also identify a condition that is sufficient to guarantee that M n ( R ) is an epsilon-crossed product, i.e. isomorphic to a crossed product associated with a unital twisted partial action. In the case where R has IBN, we provide a characterization of when M n ( R ) is an epsilon-crossed product. Our results are illustrated by several examples.

Induction from Elementary Abelian Subgroups

Let R be an associative ring which is not necessarily commutative. For any torsion theory τ on the category of left R-modules and for any nonnegative integer n we define and study the notion of the nth local cohomology functor with respect to τ. For suitably nice rings a bound for the nonvanishing of these functors is given in terms of the τ-dimension of the modules.

Let k k be any imaginary abelian field, R R the integral group ring of G = Gal ( k / Q ) G = {\text {Gal}}(k/\mathbb {Q}) , and S S the Stickelberger ideal of k k . Roughly speaking, the relative class number h − {h^ - } of k k is expressed as the index of S S in a certain ideal A A of R R described by means of G G and the complex conjugation of k ; c − h − = [ A : S ] k;{c^ - }{h^ - } = [A:S] , with a rational number c − {c^ - } in 1 2 N = { n / 2 ; n ∈ N } \frac {1} {2}\mathbb {N} = \{ n/2;n \in \mathbb {N}\} , which can be described without h − {h^ - } and is of lower than h − {h^ - } if the conductor of k k is sufficiently large (cf. [6, 9, 10]; see also [5]). We shall prove that 2 c − 2{c^ - } , a natural number, divides 2 ( [ k : Q ] / 2 ) [ k : Q ] / 2 2{([k:\mathbb {Q}]/2)^{[k:\mathbb {Q}]/2}} . In particular, if k k varies through a sequence of imaginary abelian fields of degrees bounded, then c − {c^ - } takes only a finite number of values. On the other hand, it will be shown that c − {c^ - } can take any value in 1 2 N \frac {1} {2}\mathbb {N} when k k ranges over all imaginary abelian fields. In this connection, we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields.

For a finite commutative ring R and a positive integer k ≥ 2, we construct an iteration digraph G(R, k) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = ak. In this paper, we investigate the iteration digraphs G(𝔽prCn, k) of 𝔽prCn, the group ring of a cyclic group Cnover a finite field 𝔽pr. We study the cycle structure of G(𝔽prCn, k), and explore the symmetric digraphs. Finally, we obtain necessary and sufficient conditions on 𝔽prCnand k such that G(𝔽prCn, k) is semiregular.

In this paper we generalize some well‐known commutativity theorems for associative rings as follows: Let R be a left s‐unital ring. If there exist nonnegative integers m > 1, k ≥ 0, and n ≥ 0 such that for any x, y in R, [xky − xnym, x] = 0, then R is commutative.

Let \({\sigma}\) be the class of all associative rings R with the property that if I and J are ideals of R and for all \({i \in I, j \in J}\) there exist natural numbers n, m such that i n j m = 0, then I = 0 or J = 0. We will show that \({\sigma}\) is a special class and the upper radical \({\mathcal{U}(\sigma)}\) determined by \({\sigma}\) is equal to the nil radical \({\mathcal{N}}\) of Koethe. This implies that the nil radical \({\mathcal{N}(R)}\) of any ring R is the intersection of all \({\sigma}\)-ideals I of R, that is, ideals I of R such that the factor ring R/I is in \({\sigma}\). We also give a necessary and sufficient condition for an ideal \({I \neq R}\) of a ring R to be a \({\sigma}\)-ideal.

A basis for augmentation quotients of finite abelian groups

Left dihedral codes over finite chain rings