Localization and Artinian quotient rings

The Goldie Theorem for H-semiprime algebras

“…If R is a ring with prime radical N such that Q max (R) is quasi-Frobenius with Jacobson radical J , then for Q max (R) to be a classical right quotient ring of R it is necessary and sufficient that N = J ∩ R [41].…”

Classical quotient rings

“…The next corollary was proven independently by Shock [18] (announced in [17]) and by Tachikawa [20]. Proof.…”

Orders in Artinian rings, Goldie’s Theorem and the largest left quotient ring of a ring

“…Further, for a left Noetherian ring which has a left quotient ring, Talintyre [27] has established necessary and sufficient conditions for the left quotient ring to be left Artinian. Small [21,22], Robson [20], and latter Tachikawa [25] and Hajarnavis [13] have given different criteria for a ring to have a left Artinian left quotient ring. In this paper, three more new criteria are given (Theorem 4.1, Theorem 5.1 and Theorem 6.1).…”

“…In this section we present some old criteria for a ring to have a left Artinian left quotient ring that are due to , Robson (1967), Tachikawa (1971) and Hajarnavis (1972). The starting point is Goldie's Theorem, [12], (1960) that gives an answer to the question: When does a ring have a semi-simple (Artinian) left quotient ring?…”