Commutative rings with invertible-radical factorization

Factorization in Commutative Rings

Commutative Subdirectly Irreducible Radical Rings

A ring R is radical if there is a ring S (with unit) such that R = J (S) (the Jacobson radical). We study the commutative subdirectly irreducible radical rings and show that such a ring is noetherian if and only if is finite. We present a reflection of the commutative radical rings into the category of the commutative rings and derive a lot of examples of the subdirectly irreducible radical rin...

MATH 436 Notes: Factorization in Commutative Rings

Proposition 1.1. Let f : R1 → R2 be a homomorphism of rings. If J is an ideal of R2, then f (J) is an ideal of R1 containing ker(f) and furthermore f(f(J)) ⊆ J . Now let f : R1 → R2 be an epimorphism of rings. If J is an ideal of R2 then f(f (J)) = J . If I is an ideal of R1 then f(I) is an ideal of R2. Furthermore we have I ⊆ f (f(I)) = I + ker(f) and thus I = f(f(I)) if I contains ker(f). Thu...

Tiling with Commutative Rings

so that every square is covered by exactly one domino? In other words, can R be tiled by vertical and horizontal dominoes? The coloring gives the answer to this well-known problem away. The region R has 32 black squares and 30 white squares. Since each domino covers exactly one black and one white square, no tiling is possible. The aim of this article is to explain a way to tackle tiling proble...

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...