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Prime ideals in universal algebras. (English) Zbl 0554.08002
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
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