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T-spaces in associative algebras. (English. Russian original) Zbl 1153.16020

J. Math. Sci., New York 143, No. 5, 3451-3508 (2007); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 120 (2005).
Let \(K\) be a fixed Noetherian commutative (associative) ring with 1. The paper under review studies T-spaces in free algebras over \(K\). During the last decade it turned out that one may study T-spaces in order to obtain information concerning the corresponding T-ideals. Recall that a T-space in the free associative algebra \(K(X)\) is a \(K\)-submodule that is closed under endomorphisms. For example the set of all central polynomials for a given algebra (including its identities) forms a T-space. In fact the resemblance with T-ideals is not complete: there exist T-spaces that do not coincide with the central polynomials of any algebra. Nevertheless T-spaces proved to be extremely useful in the study of the finite basis properties for T-ideals and related questions.
The paper under review deals with such problems concerning T-spaces. It is proved that every T-space in the \(K\)-algebra of the commutative (and associative) polynomials is finitely generated as a T-space. Recall that if \(K\) is an infinite field of positive characteristic there exist T-spaces of the free associative algebra (even containing the polynomial \([[x_1,x_2],x_3]\)) that are not finitely generated, see for more details the survey by A. V. Grishin and V. V. Shchigolev [J. Math. Sci., New York 134, No. 1, 1799-1878 (2006); translation from Sovrem. Mat. Prilozh. 18, 26-97 (2004; Zbl 1100.16018)].
Furthermore, the author studies the variety of algebras \(V(n)\) defined by the identity \[ [x_1,x_2]\cdots[x_{2n-1},x_{2n}]. \] One defines restricted T-spaces as T-spaces that no generator contains \(x^{N+1}\) for each \(x\in X\). The main result here is that all restricted T-spaces in free algebras of \(V(n)\) are finitely generated. Then it is proved that the T-spaces in the free algebra of the variety \(V(n,N)\) determined by the above product of commutators and by \(x^N=0\), are finitely generated.
The most interesting results concern the so-called limit (or just-nonfinitely based) T-spaces. These are T-spaces that are not finitely generated but all larger T-spaces are finitely generated. (Analogously one defines limit T-ideals.) Recall that if there exists a nonfinitely generated T-space by the Zorn Lemma there must exist limit ones. Let now \(K\) be a field of characteristic \(p>0\), and define \(V_p\) to be the variety of algebras defined by \([[x_1,x_2],x_3]\) and by \(x^4\) if \(p=2\), and by \(x^p\) if \(p>2\). It is shown that \(V_p\) is a minimal variety with the property that its free algebras contain non-finitely generated T-spaces. (When \(p>2\) the variety \(V_p\) is generated by the infinite dimensional Grassmann algebra.) Recall that the similar problem for algebras is still open. Several more examples of limit T-spaces are obtained.
The paper surveys a couple of papers published originally in Russian. It is an excellent idea to have them in English readily available to the specialists in the area.

MSC:

16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings
16R50 Other kinds of identities (generalized polynomial, rational, involution)

Citations:

Zbl 1100.16018
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References:

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