Subdirect decomposition of rings and the axiom of choice
It is shown that the following is a theorem of ZF set theory without choice: A commutative ring R admits a representation as a subdirect product of subdirectly irreducible rings if and only if R contains a maximal ideal. It follows then from a theorem of \textit{W. Hodges} [J. Lond. Math. Soc., II. Ser. 19, 285-287 (1979; Zbl 0403.03037)] that already for the class of unique factorization domains, the subdirect product representability by subdirect irreducibles implies the axiom of choice. This answers a question of \textit{G. Grätzer} [Universal algebra, 2nd edition (1979; Zbl 0412.08001), p. 161, Problem 30]. We also draw attention to the little known fact that the very notion and terminology of subdirect product representation by subdirect irreducibles were invented by \textit{R. Remak} in 1930 [cf. J. Reine Angew. Math. 163, 1-44 (1930); Jahrb. Fortschr. Math. 56, I, p. 129].
scientific article; zbMATH DE number 4021130