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...

We classify all the simple modules for the algebra of relations on a finite set, give their dimension, and find the dimension of the Jacobson radical of the algebra. AMS Subject Classification : 06A06, 13C05, 16B50, 16D60, 16N20, 18B05, 18B10, 18B35, 18E05.

We introduce a set that is tightly close to the set of the Jacobson radical of module (the intersection of all maximal elements in support). In the last section, it is proved that the set of zero divisors of a module is equal to the union of the maximal elements of the support of module if the module is finitely generated and injective.

In this paper we survey some results on the structure of noncommutative rings. We focus particularly on nil rings, Jacobson radical rings and rings with finite Gelfand–Kirillov dimension. Mathematics Subject Classification (2000). 16-02, 16-06, 16N40, 16N20, 16N60, 16D60, 16P90.

Let R be a ring with center Z, Jacobson radical J , and set N of all nilpotent elements. Call R semiperiodic if for each x ∈ R\ (J ∪Z), there exist positive integers m, n of opposite parity such that x − x ∈ N . We investigate commutativity of semiperiodic rings, and we provide noncommutative examples. Mathematics Subject Classification (2000). 16U80.

We present a practical algorithm to construct the subgroup of the general linear group that preserves a system of bilinear or sesquilinear forms on a vector space defined over a finite field. Components include efficient algorithms to construct the Jacobson radical and the group of units of a matrix algebra.

Those linearly compact rings with identity having a simple group of units and a transsnitely nilpotent Jacobson radical are identiied. A consequence of this characterization is Cohen and Koh's classiication theorem for compact rings with identity having a simple group of units.

We investigate the computational complexity of deciding whether or not a given polynomial , presented as the sum of monomials, is identically 0 over a ring. It is proved that if the factor by the Jacobson-radical is not commutative, then the problem is coNP-complete.