Classical Radicals of Invariants of Algebraic Skew Derivations

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

The soft Jacobson radical of a commutative ring

In this paper, the notion of the soft Jacobson radical of a ring is defined. A relationship between the soft Jacobson radical of a ring and Jacobson semisimple ring is established. Some properties of this notion have been studied under homomorphism.

Comaximal graph of amalgamated algebras along an ideal

Let [Formula: see text] and [Formula: see text] be commutative rings with identity, [Formula: see text] be an ideal of [Formula: see text], and let [Formula: see text] be a ring homomorphism. The amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text] denoted by [Formula: see text] was introduced by D’Anna et al. in 2010. In this paper, we investigate some properties of the comaximal graph of [Formula: see text] which are transferred to the comaximal graph of [Formula: see text], and also we study some algebraic properties of the ring [Formula: see text] by way of graph theory. The comaximal graph of [Formula: see text], [Formula: see text], was introduced by Sharma and Bhatwadekar in 1995. The vertices of [Formula: see text] are all elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be the subgraph of [Formula: see text] generated by non-unit elements, and let [Formula: see text] be the Jacobson radical of [Formula: see text]. It is shown that the diameter of the graph [Formula: see text] is equal to the diameter of the graph [Formula: see text], and the girth of the graph [Formula: see text] is equal to the girth of the graph [Formula: see text], provided some special conditions.

An Equivalent Condition and Some Properties of Strong J-Symmetric Ring

Let J R denote the Jacobson radical of a ring R . We say that ring R is strong J-symmetric if, for any a , b , c ∈ R , a b c ∈ J R implies b a c ∈ J R . If ring R is strong J-symmetric, then it is proved that R x / x n is strong J-symmetric for any n ≥ 2 . If R and S are rings and W S R is a R , S -bimodule, E = T R , S , W = R W 0 S = r w 0 s | r ∈ R , w ∈ W , s ∈ S , then it is proved that R and S are J-symmetric if and only if E is J-symmetric. It is also proved that R and S are strong J-symmetric if and only if E is strong J-symmetric.

Abelian p-groups with minimal full inertia

The class of abelian p-groups with minimal full inertia, that is, satisfying the property that fully inert subgroups are commensurable with fully invariant subgroups is investigated, as well as the class of groups not satisfying this property; it is known that both the class of direct sums of cyclic groups and that of torsion-complete groups are of the first type. It is proved that groups with “small" endomorphism ring do not satisfy the property and concrete examples of them are provided via Corner’s realization theorems. Closure properties with respect to direct sums of the two classes of groups are also studied. A topological condition of the socle and a structural condition of the Jacobson radical of the endomorphism ring of a p-group G, both of which are satisfied by direct sums of cyclic groups and by torsion-complete groups, are shown to be independent of the property of having minimal full inertia. The new examples of fully inert subgroups, which are proved not to be commensurable with fully invariant subgroups, are shown not to be uniformly fully inert.

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings. The properties of radicals of pseudo-reduced-over-center rings are investigated, especially related to polynomial rings. It is proved that for pseudo-reduced-over-center rings of nonzero characteristic, the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals. For a locally finite ring [Formula: see text], it is proved that if [Formula: see text] is pseudo-reduced-over-center, then [Formula: see text] is commutative and [Formula: see text] is a commutative regular ring with [Formula: see text] nil, where [Formula: see text] is the Jacobson radical of [Formula: see text].