On the Jacobson Radical of Semigroup Graded Rings

Semigroup-graded rings with finite support

“…The class of Jacobson rings is a left and right hereditary radical class [9], [18] from which fact analogues of properties (ii), (iv), and (v) are immediate. The other properties of Jacobson rings are verified in [2], [4], [16]. (Note that [4] proves (iii) for Jacobson rings in the case that R is a direct sum of a finite set of its right ideals: such a ring is graded by a left zero .semigroup with the ideals as homogeneous components.…”

“…In particular, we have the following result. Verifications of the appropriate closure properties which are not straight-forward can be found in [1], [2], [3] for semilocal and semiprimary rings, in [6] for nilpotent rings, and in [11], [17] for Pi-rings. The class of Jacobson rings is a left and right hereditary radical class [9], [18] from which fact analogues of properties (ii), (iv), and (v) are immediate.…”

“…When S is finite, an approach based on the structure theory of semigroups has led to an affirmative answer to this question for perfect, semilocal and semiprimary rings [3], for Jacobson rings [4], and for Pi-rings [11].…”

“…This approach leads to difficulties because the structure of such factors is not well understood. In the absence of such factors, the proofs developed for finite semigroups can be easily modified to handle the finite support case [2].…”

“…If this were always possible, then the methods of [2], [3] denote the right and two-sided ideals generated by A. Note that a oneor two-sided ideal generated by homogeneous elements is necessarily homogeneous.…”