Abstract
Let F = k{X} be the free associative algebra over a field k, X an infinite countable set. It is well-known that the set of polynomial identities I(R) of a given k-algebra R is a T-ideal, namely an ideal of F which is invariant under all the algebra endomorphisms of F. Moreover, if J is a T-ideal then J = I(F/ J) [3, p. 61].
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References
Cohn, P. M. Universal Algebra (2nd ed.), D. Reidel, Dordrecht, 1981.
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Klein, A. A. Necessary conditions for embedding rings into fields, Trans. Amer. Math. Soc. 137 (1969), 141–151.
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Klein, A.A. (1990). The Set of Quasi-Identities of an Algebra. In: Almeida, J., Bordalo, G., Dwinger, P. (eds) Lattices, Semigroups, and Universal Algebra. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-2608-1_15
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DOI: https://doi.org/10.1007/978-1-4899-2608-1_15
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