Munn [11] proved that the Jacobson radical of a commutative semigroup ring is nil provided that the radical of the coefficient ring is nil. This was generalized, for semigroup algebras satisfying polynomial identities, by Okniński [14] (cf. [15, Chapter 21]), and for semigroup rings of commutative semigroups with Noetherian rings of coefficients, by Jespers [4]. It would be interesting to obtai...

For a ring endomorphism α and an α-derivation δ, we introduce α-compatible ideals which are a generalization of α-rigid ideals and study the connections of the prime radical and the upper nil radical of R with the prime radical and the upper nil radical of the Ore extension R[x;α, δ] and the skew power series R[[x;α]]. As a consequence we obtain a generalization of Hong, Kwak and Rizvi, 2005.

It is proved that a ring is periodic if and only if, for any elements x and y , there exist positive integers k,l,m, and n with either k =m or l =n, depending on x and y , for which xkyl = xmyn. Necessary and sufficient conditions are established for a ring to be a direct sum of a nil ring and a J-ring. 2000 Mathematics Subject Classification. Primary 16U99, 16N20, 16D70.

A Boolean ring satisfies the identity x2 = x which, of course, implies the identity x2y − xy2 = 0. With this as motivation, we define a subBoolean ring to be a ring R which satisfies the condition that x2y−xy2 is nilpotent for certain elements x, y of R. We consider some conditions which imply that the subBoolean ring R is commutative or has a nil commutator ideal. Mathematics Subject Classific...

For a commutative ring R, let Nil(R) be the set of all nilpotent elements of R, Z(R) be the set of all zero divisors of R, T (R) be the total quotient ring of R, and H = {R | R is a commutative ring and Nil(R) is a divided prime ideal of R}. For a ring R ∈ H, let φ : T (R) −→ RNil(R) such that φ(a/b) = a/b for every a ∈ R and b ∈ R\Z(R). A ring R is called a ZPUI ring if every proper ideal of R...

We define the notions of finite-state and functionally recursive matrices and their growth. We also introduce two rings generated by functionally recursive matrices. The first is isomorphic to the 2-generated free ring. The second is a 2-generated monomial ring such that the multiplicative semigroup of monomials in the generators is nil of degree 5 and the ring has Gelfand Kirillov dimension 1 ...

The notions of a right quasiregular element and right modular right ideal in a near-ring are initiated. Based on these J 0(R), the right Jacobson radical of type-0 of a near-ring R is introduced. It is obtained that J 0 is a radical map andN(R)⊆ J 0(R), whereN(R) is the nil radical of a near-ring R. Some characterizations of J 0(R) are given and its relation with some of the radicals is also di...

Let G be a semi-simple simply-connected complex algebraic group and T ⊂ B a maximal torus and a Borel subgroup respectively. Let h = Lie T be the Cartan subalgebra of the Lie algebra Lie G, and W := N(T )/T the Weyl group associated to the pair (G, T ), where N(T ) is the normalizer of T in G. We can view any element w = w mod T ∈ W as the element (denoted by the corresponding German character)...