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Removed deprecated (abstract-algebra) tag - see the tag info: https://mathoverflow.net/tags/abstract-algebra/info (if there are some other suitable tags, choose them instead.)

edited Oct 9, 2017 at 23:46

Martin Sleziak

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What are algebraic systems and algebraic closure as defined by Kenjiro Shoda? Which are his main results on them?

In On Utumi's ring of quotients, Canad. J. Math. 15(1963), 363-370, J. Lambek says:

As a matter of historical record, the minimal injective extension of a module is a special case of the "algebraic closure" of an algebraic system considered by K. Shoda in his paper Zur Théorie der algebraischen Erweiterungen, Osaka Math. J., 4 (1952), 133-144.

Since I cannot read German, I cannot understand what the paper by Shoda says, although I think that he does defer the definitions to his previous papers Uber die allgemeinen algebraischen Systeme I-VIII (links provided below).

I would like to know:

What exactly his algebraic systems are (maybe the varieties of universal algebra?).
Which are his fundamental results about said systems (in these or other papers).

A published translation/review/summary would serve me perfectly. The MathSciNet reviews of the first papers are too vague, enumerating examples and talking about "certain operations", "various relations", etc.

Links to the I-VIII papers:

ra.rings-and-algebras

asked Oct 9, 2017 at 19:22

Jose Brox