Graded radicals of graded rings

Radical Theory for Granded Rings

“…We remark that the correspondence 31 -• 3l G is one-to-one, order preserving and preserves lattice meets since if 31 ^ and 3l 2 (3£(A) H is a graded semisimple class. Now suppose that 31 is a radical semisimple class of associative rings (see [7]).…”

“…For each A e A define /»/ = {a e ^|ap^ e P A for all g e G } . As in [2], the P\ are graded prime ideals of A and since />/#(?• C P A for all X 6 A, n{P/: A e A} = 0. For each A e A, …”

“…PROPOSITION We omit the straightforward proof of this proposition. The graded prime radical class, the graded strongly prime radical class of [10] and, when G is finite, the graded Jacobson radical class of [5] (see [6, Theorem 4.4(1)]) are radical classes of this kind, as are the graded Levitzki and graded von Neumann regular radical classes of [2].…”

“…Reflected radical classes were introduced in [2] where graded rings were studied in the category which contained only those homomorphisms of degree terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700034315 (e, e).…”

A radical for graded Lie algebras

“…The prime radical, also known as the Baer radical, is defined as the intersection of all prime ideals of a ring and is the least ideal giving semiprime quotient. Its graded version is the so called graded-prime radical of graded associative rings and algebras, and was investigated in works such as [1], [2], [8]. In [2] this radical is defined as the intersection of all graded-prime ideals, it is the least giving graded-semiprime quotient, and coincides with the largest graded ideal contained in the usual prime radical, see [2, 5.1].…”