Some results on coherent rings

Nil＊-COHERENT RINGS

“…By assumption, M is a finitely generated submodule of the left R-module N (S), and hence M is a finitely presented left R-module. So M is a finitely presented left S-module by [11,Theorem 1]. Therefore, S is a left N il * -coherent ring.…”

“…The ring R is said to be left coherent if R is a coherent as a left R-module. Since coherence of rings and modules first appeared in [2], their generalizations have been studied extensively by many authors (see, [3,4,7,9,11,15,17]). A ring R is called left J-coherent [7] if the Jacobson radical J(R) of R is a coherent left R-module.…”

Some results on coherent rings II

“…The result stated in [1, p. 63, Exercise 12a] is a basic theorem of [2] and first appeared there. A variety of results on and examples of coherent rings and modules are presented in [3].In this note, all rings contain an identity, all modules are unitary, and all ring homomorphisms " preserve " identities. If the underlying ring is non-commutative, all definitions and results will be given for the left side; the " right side " case will be immediate.…”

“…The result stated in [1, p. 63, Exercise 12a] is a basic theorem of [2] and first appeared there. A variety of results on and examples of coherent rings and modules are presented in [3].…”

“…A result of [3] states that, if S is a multiplicative system in the commutative coherent ring A, then A s must also be coherent. Here we show that, if every localization at a maximal ideal of a semi-local ring is coherent, then A is also coherent.…”