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 rings with various properties. At last, we show partial results in the classification of the factors R/M of the subdirectly irreducible radical rings R by their monoliths M . For a ring R we denote J (R) = ⋂ {AnnR(M)| M is a simple R-module} the Jacobson’s radical of R. Radical rings are just all Jacobson’s radicals of all rings. These rings are important not only for this property, but were also massively used, together with their adjoint groups, by E. Zelmanov in solving of the Burnside’s problem for finitely generated groups. Equivalently, a ring R is radical if and only if for every a ∈ R there is an adjoint element ã ∈ R such that a+ ã+ aã = 0. Thus we can view the class of the radical rings as a universal algebraic variety (with one nullary, two unary and two binary operations). Since every simple radical ring is isomorphic to a zero-multiplication ring Zp for a prime p, we proceed naturally by investigating of the structure of this variety to the subdirectly irreducible ones. A subdirectly irreducible ring is one in which the intersection of all the nonzero ideals is a nonzero ideal. These ring are a kind of building blocks, since, by the Birkhoff’s theorem, every radical ring is isomorphic to a subdirect product of subdirectly irreducible radical rings. In this paper we pay our attention on the commutative rings. Subdirectly irreducible commutative ring were already studied by N.H. McCoy [6] and N. Divinsky [7]. In [6] was shown that these rings are of the following three types: (α) Fields. (β) Every element is a zero divisor. (γ) There exists both non-divisors of zero and nilpotent elements. The subdirectly irreducible commutative radical rings are of type (β). In addition, by [6], if they satisfy either the descending or the ascending condition, they are nilpotent. An important property of the class of commutative radical rings is also the existence of a reflection of the category of the commutative rings into the category of the commutative radical rings. In this paper we present such reflection, which will be consecutively a very effective tool for constructing of examples of subdirectly irreducible radical rings with various properties. As we will see, a very helpful class for these constructions will be the class of so called subradical rings. 2000 Mathematics Subject Classification. 16D90, 16E30 (primary), 03E75, 18G25, 20K40, 16D40, 16E05, 16G99 (secondary).