The class of rings $\mathcal{J}=\{A (A,\circ)$ forms a group$\}$ radical and it is called the Jacobson class. For any ring $A$, $\mathcal{J}(A)$ $A$ defined as largest ideal which belongs to $\mathcal{J}$. In fact, one most important classes since used widely in another branch abstract algebra, for example, construct two-sided brace. On other hand, every Morita context $T=\begin{pmatrix} R & V ...

This article expands upon the recent work by Downey, Lempp, and Mileti [3], who classified the complexity of the nilradical and Jacobson radical of commutative rings in terms of the arithmetical hierarchy. Let R be a computable (not necessarily commutative) ring with identity. Then it follows from the definitions that the prime radical of R is Π1, and the Levitzki radical of R is Π 0 2. We show...

Let G = Cm p o C2 be a generalized dihedral group for an odd prime and natural number m, L M(G; 2) the RA2 loop obtained from F finite field of characteristic 2. For algebra F[L], we determine Jacobson radical J(F[L]) F[L] Wedderburn decomposition F[L]=J(F[L]). The structure 1 + is also determined.

Let [Formula: see text] be a commutative ring with identity. The co-maximal ideal graph of text], denoted by is simple whose vertices are proper ideals which not contained in the Jacobson radical and two distinct adjacent if only text]. In this paper, we use Gallai’s Theorem concept strong resolving to compute metric dimension for graphs rings. Explicit formulae dimension, depending on whether ...

let  be a banach algebra and  a derivation. in this paper, it is proved, under certain conditions, that , where  is the jacobson radical of . moreover, we prove that if  is unital and  is a continuous derivation, then , where  denotes the set of all primitive ideals such that  is commutative,  denotes the set of all maximal (modular) ideals such that  is commutative, and  is the set of all non-...

It is a famous conjecture that every derivation on each Banach algebra leaves every primitive ideal of the algebra invariant. This conjecture is known to be true if, in addition, the derivation is assumed to be continuous. It is also known to be true if the algebra is commutative, in which case the derivation necessarily maps into the (Jacobson) radical. Because I. M. Singer and J. Wermer origi...

where (a) denotes the two-sided ideal of R generated by a. Then J is the Jacobson radical [6 ] of R, and N is the radical of R as defined in [l]. It is well known that J = 0 if and only if R is isomorphic to a subdirect sum of primitive rings, and ^ = 0 if and only if R is isomorphic to a subdirect sum of simple rings with unit element. The above definitions of / and N suggest that it might be ...