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Associative generalized rings. (English) Zbl 0719.16027
A generalized ring is a triple \((G,+,\Omega)\), where (i) \((G,+)\) is an Abelian group and (ii) \(\Omega\) is a family of finitary operations on G such that each \(\omega\in \Omega\) is distributive, i.e. \[ \omega (g_ 1,...,g_{i-1},g_ i+g,g_{i+1},...,g_ n)=\omega (g_ 1,...,g_{i- 1},g_ i,g_{i+1},...,g_ n)+\omega (g_ 1,...,g_{i- 1},g,g_{i+1},...,g_ n),\text{ for all } g_ 1,...,g_ n,\quad g\in G. \] Ideals, commutativity and direct products are defined in the natural way. A notion of distributivity is also defined. It is shown that the direct product of a family of commutative (associative) generalized rings is again a commutative (associative) generalized ring. Principal ideals are characterized, and various results are obtained concerning the ideals of an associative commutative generalized ring (ACGR).
Reviewer’s remark: The definition of generalized rings includes rings, non-associative rings, ternary rings [W. G. Lister, Trans. Am. Math. Soc. 154, 37-55 (1971; Zbl 0216.069)], \(\Gamma\)-rings [W. E. Barnes, Pac. J. Math. 18, No.3, 411-422 (1966; Zbl 0161.033)] and R- modules, but not near-rings. The definition of commutativity coincides with the accepted one in each of these varieties. Rings and ternary rings are associative in this sense, but R-modules and \(\Gamma\)-rings are not.
16Y99 Generalizations
16D25 Ideals in associative algebras
17A40 Ternary compositions
08A62 Finitary algebras