We study symmetric algebras $A$ over a field $F$ in which the Jacobson radical of center $A$, socle or Reynolds ideal are ideals.

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

Given a semigroup S, we prove that if the upper nilradical Nil∗(R) is homogeneous whenever R is an S-graded ring, then the semigroup S must be cancelative and torsion-free. In case S is commutative the converse is true. Analogs of these results are established for other radicals and ideals. We also describe a large class of semigroups S with the property that whenever R is a Jacobson radical ri...

We describe the representation theory of finitely generated indecomposable modules over artin algebras which do not lie on cycles involving homomorphisms from infinite Jacobson radical module category.