Infinite subrings of infinite rings and near-rings
The major theorem of the paper establishes a class \(I_0\) of infinite commutative associative rings with the property that every infinite alternative ring contains a subring belonging to \(I_0\). As a consequence of this theorem and some of the machinery of its proof, it is shown that an alternative ring \(R\) is finite if it satisfies one of the following: (1) \(R\) has both a.c.c. and d.c.c. on associative subrings. (2) \(R\) has only a finite number of zero subrings and a finite number of subrings which are integral domains. (3) \(R\) has nonzero divisors of zero and the two-sided annihilator of each non-zero two-sided zero divisor is finite. (4) \(R\) has nonzero divisors of zero and satisfies a.c.c. and d.c.c. on subrings consisting of two-sided zero-divisors of \(R\). In one section of the paper some consideration is given to the question of the sense in which \(I_0\) is minimal. In the final section, the description of \(I_0\) is employed in a proof that an infinite distributive near-ring \(R\) must contain an infinite subring unless the derived group \(R'\) of \(R^+\) is an infinite group containing no infinite abelian subgroup.
scientific article; zbMATH DE number 3470627