We establish a correspondence between abelian regular subgroup of the affine group, and commutative, associative algebra structures on the underlying vector space that are (Jacobson) radical rings. As an application, we show that if the underlying field has positive characteristic, then an abelian regular subgroup has finite exponent if the vector space is finite-dimensional, while it can be to...

Abstract The class of abelian p -groups with minimal full inertia, that is, satisfying the property fully inert subgroups are commensurable invariant is investigated, as well groups not this property; it known both direct sums cyclic and torsion-complete first type. It proved “small" endomorphism ring do satisfy concrete examples them provided via Corner’s realization theorems. Closure properti...

Abstract We show that a direct limit of projective contramodules (over right linear topological ring) is if it has cover. A similar result obtained for $\infty $-strictly flat dimension not exceeding $1$, using an argument based on the notion Jacobson radical. Covers and precovers limits more general classes objects, both in abelian categories with exact nonexact limits, are also discussed, eye...

A ring R is defined to be semiabelian if every idempotent of R is either right semicentral or left semicentral. It is proved that the set N(R) of nilpotent elements in a π-regular ring R is an ideal of R if and only if R/J(R) is abelian, where J(R) is the Jacobson radical of R. It follows that a semiabelian ring R is π-regular if and only if N(R) is an ideal of R and R/N(R) is regular, which ex...

A ring Λ is called a tiled order if Λ is a prime Noetherian semi-prefect semi-distributive ring with non-zero Jacobson radical. Any tiled order can be constructed from a (non-necessarily commutative) discrete valuation ring and an exponent matrix. The latter means a square integer matrix A = (αps), whose diagonal entries are equal to zero, and for all possible indices i, j, k, the following ine...

In this paper we provide various properties of Rad-⊕-supplemented modules. In particular, we prove that a projective module M is Rad⊕-supplemented if and only if M is ⊕-supplemented, and then we show that a commutative ring R is an artinian serial ring if and only if every left R-module is Rad-⊕-supplemented. Moreover, every left R-module has the property (P ∗) if and only if R is an artinian s...

1. W. Ambrose, Structure theorem for a special class of Banach algebras, Trans. Amer. Math. Soc. vol. 57 (1945) pp. 364-386. 2. N. Jacobson, The radical and semi-simplicity for arbitrary rings, Amer. J. Math, vol. 67 (1945) pp. 300-320. 3. I. Kaplansky, Dual rings, Ann. of Math. vol. 49 (1948) pp. 689-701. 4. L. H. Loomis, An introduction to harmonic analysis, New York, Van Nostrand, 1953, pp. ...

A ghost in the stable module category of a group G is a map between representations of G that is invisible to Tate cohomology. We show that the only non-trivial finite p-groups whose stable module categories have no non-trivial ghosts are the cyclic groups C2 and C3. We compare this to the situation in the derived category of a commutative ring. We also determine for which groups G the second p...