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Suppose that T is an equational theory of rings. If T is finitely based, then there is a least number m such that T can be defined by m equations. This number m can depend on the operation symbols that occur in T. In the late 1960s, Alfred Tarski and Thomas Green completely determined the values of m for arbitrary equational theories of rings. These results were announced by Tarski in his now famous seminal survey article published in 1968. No proofs were mentioned in this paper. In the meantime, various authors independently proved several results obtained by Tarski and Green. In this talk, I would like to present a brief account of their discovery (in rings) and their proofs about the minimal bases for rings (obtained by Gratzer, Padmanabhan, Barry Wolk). For a relatively complete description of their results in groups and rings, please see the recent survey article by George McNulty.

I have long forgotten this topic. However, a recent e-mail from Tommy Kucera on "group axioms" made me think about the minimal equational bases once again. Being a seminar on Rings and Modules, I thought I should share this with other members of the Seminar. We will resume our discussion of "Taxi Cab Numbers and the Rank" next time.

Refences
A. Tarski, Equational logic and equational theories of algebras, in: Contributions to Mathematical Logic (Colloquium, Hannover, 1966), North-Holland, Amsterdam, 1968, pp. 275-288.
G. Grätzer and R. Padmanabhan, Symmetric difference in abelian groups. Pacific J. Math. 74 (1978), pp. 339-347.
R. Padmanabhan and B. Wolk, Equational theories with a minority polynomial. Proc. Amer. Math. Soc. 83 (1981), pp. 238-242.
McNulty, George F. Minimum bases for equational theories of groups and rings: the work of Alfred Tarski and Thomas Green. Provinces of logic determined. Ann. Pure Appl. Logic 127 (2004).