On Generalized Probability in Finite Commutative Rings

Finite Commutative Rings

‡ Every function on a finite residue class ring D/I of a Dedekind domain D is induced by an integer-valued polynomial on D that preserves congruences mod I if and only if I is a power of a prime ideal. If R is a finite commutative local ring with maximal ideal P of nilpotency N satisfying for all a, b∈R, if ab∈ P then a∈ P k , b∈ P j with k+ j ≥min(n,N), we determine the number of functions (as...

Polynomial functions on finite commutative rings

Every function on a nite residue class ring D=I of a Dedekind domain D is induced by an integer-valued polynomial on D that preserves congruences mod I if and only if I is a power of a prime ideal. If R is a nite commutative local ring with maximal ideal P of nilpotency N satisfying for all a; b 2 R, if ab 2 Pn then a 2 P k , b 2 P j with k + j min(n;N), we determine the number of functions (as...

Commutative Rings and Finite Fields

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...

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.