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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Product of ring varieties and attainability


Author: Awad A. Iskander
Journal: Trans. Amer. Math. Soc. 193 (1974), 231-238
MSC: Primary 16A48
MathSciNet review: 0349753
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Abstract: The class of all rings that are Everett extensions of a ring in a variety $ \mathfrak{U}$ by a ring in a variety $ \mathfrak{B}$ is a variety $ \mathfrak{U} \cdot \mathfrak{B}$. With respect to this operation the set of all ring varieties is a partially ordered groupoid (under inclusion), that is not associative. A variety is idempotent iff it is the variety of all rings, or generated by a finite number of finite fields. No families of polynomial identities other than those equivalent to $ x = x$ or $ x = y$ are attainable on the class of all rings or on the class of all commutative rings.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0349753-0
Keywords: Associative ring, variety, free ring, T-ideal, polynomial identity, indecomposable, attainable set of identities, Everett extension
Article copyright: © Copyright 1974 American Mathematical Society