Basic theorems on general commutative rings

Approximation theorems for valuations on commutative rings

ON COMMUTATIVE GELFAND RINGS

A ring is called a Gelfand ring (pm ring ) if each prime ideal is contained in a unique maximal ideal. For a Gelfand ring R with Jacobson radical zero, we show that the following are equivalent: (1) R is Artinian; (2) R is Noetherian; (3) R has a finite Goldie dimension; (4) Every maximal ideal is generated by an idempotent; (5) Max (R) is finite. We also give the following resu1ts:an ideal...

Basic Subgroups in Commutative Modular Group Rings

Let S(RG) be a normed Sylow p-subgroup in a group ring RG of an abelian group G with p-component Gp and a p-basic subgroup B over a commutative unitary ring R with prime characteristic p. The first central result is that 1 + I(RG;Bp) + I(R(p)G;G) is basic in S(RG) and B[1 + I(RG;Bp) + I(R(p )G;G)] is p-basic in V (RG), and [1 + I(RG;Bp) + I(R(p )G;G)]Gp/Gp is basic in S(RG)/Gp and [1 + I(RG;Bp)...

On Commutative Reduced Baer Rings

It is shown that a commutative reduced ring R is a Baer ring if and only if it is a CS-ring; if and only if every dense subset of Spec (R) containing Max (R) is an extremally disconnected space; if and only if every non-zero ideal of R is essential in a principal ideal generated by an idempotent.

on commutative reduced baer rings

it is shown that a commutative reduced ring r is a baer ring if and only if it is a cs-ring; if and only if every dense subset of spec (r) containing max (r) is an extremally disconnected space; if and only if every non-zero ideal of r is essential in a principal ideal generated by an idempotent.