Prime ideals in universal algebras. (English) Zbl 0554.08002

Ursini, Aldo

Acta Univ. Carol., Math. Phys. 25, No. 1, 75-87 (1984).

Let V be a variety with a constant 0. A term \(p(x_ 1,...,x_ m,y_ 1,...,y_ n)\) is an ideal term, if p(\(\vec x,\vec y)=0\) is an equation in V. A subset I of \(A\in V\) is an ideal iff \(p(A^ m,I^ n)\subseteq I\) for every ideal term. The author assumes that every ideal is the 0- class of a unique congruence relation. Consequently the variety must be congruence modular, so the commutator multiplication is used to define ideal multiplication. In this setting, however, alternative descriptions of \(I\cdot J\) are available. In particular prime ideals and the prime radical can be defined and basic results are shown.
The author then heads for a generalization of I. S. Cohen’s result for commutative rings with identity, stating that every ideal is finitely generated, provided all prime ideals are. Naturally, in this abstract setting, a number of hypotheses which are automatically true in commutative rings with identity, have to be added. For the proof the author introduces A-\(\sigma\)-complexes, generalizing the notion of module in the ring case.

Reviewer: H.-P.Gumm

MSC:

08A05	Structure theory of algebraic structures
08B10	Congruence modularity, congruence distributivity
08B05	Equational logic, Mal’tsev conditions
13A15	Ideals and multiplicative ideal theory in commutative rings

Keywords:

congruence relation; congruence modular; commutator multiplication; prime ideals; prime radical