×

Graded identities and PI equivalence of algebras in positive characteristic. (English) Zbl 1077.16024

The building blocks of the structure theory of T-ideals over a field \(K\) of characteristic 0 developed by Kemer are the matrix algebras \(M_n(K)\) and \(M_n(E)\) with entries, respectively, from \(K\) and from the Grassmann algebra \(E\), and the algebras \(M_{a,b}(E)\) which are subalgebras of \(M_{a+b}(E)\). Up to PI-equivalence, the set of these algebras is closed under tensor products. In particular, \(T(M_{1,1}(E))=T(E\otimes E)\) and \(T(M_{a,b}(E)\otimes E)=T(M_{a+b}(E))\).
Recently, the authors showed that the first PI-equivalence does not hold over an infinite field of positive characteristic \(p>2\) [J. Algebra 276, No. 2, 836-845 (2004; Zbl 1065.16017)]. In the present paper they consider again the case of characteristic \(p>2\). They find bases of the \(G\)-graded polynomial identities of the algebras \(M_n(E)\) and \(M_{a,b}(E)\otimes E\), where \(n=a+b\) and \(G=\mathbb{Z}_n\times\mathbb{Z}_2\). This extends to characteristic \(p\) a result of O. M. Di Vincenzo and V. Nardozza [Commun. Algebra 31, No. 3, 1453-1474 (2003; Zbl 1039.16023)].
In particular, the algebra \(M_{a,a}(E)\otimes E\) has more graded identities than the algebra \(M_{2a}(E)\). This allows to show that \(M_{1,1}(E)\otimes E\) satisfies an ordinary polynomial identity which does not hold for \(M_2(E)\), in positive characteristic \(p>2\).

MSC:

16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16R50 Other kinds of identities (generalized polynomial, rational, involution)
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
15A75 Exterior algebra, Grassmann algebras
16W50 Graded rings and modules (associative rings and algebras)
16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings
16S50 Endomorphism rings; matrix rings
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1081/AGB-120016017 · Zbl 1015.16024 · doi:10.1081/AGB-120016017
[2] Azevedo S. S., Serdica 29 pp 149– (2003)
[3] DOI: 10.1016/j.jalgebra.2004.01.004 · Zbl 1065.16017 · doi:10.1016/j.jalgebra.2004.01.004
[4] Bahturin Yu., Alg. (Moscow) pp 57– (1998)
[5] DOI: 10.1080/00927879308824632 · Zbl 0816.16025 · doi:10.1080/00927879308824632
[6] Chiripov P., Pliska Stud. Math. Bulgar. 2 pp 103– (1981)
[7] DOI: 10.1016/j.laa.2003.07.011 · Zbl 1044.16016 · doi:10.1016/j.laa.2003.07.011
[8] DOI: 10.1007/BF02808074 · Zbl 0784.16015 · doi:10.1007/BF02808074
[9] DOI: 10.1081/AGB-120017775 · Zbl 1039.16023 · doi:10.1081/AGB-120017775
[10] Di Vincenzo O. M., Rend. Sem. Mat. Univ. Padova 108 pp 27– (2002)
[11] DOI: 10.1007/BF01669018 · Zbl 0496.16017 · doi:10.1007/BF01669018
[12] Kemer A., Translations Math. Monographs 87 (1991)
[13] Koshlukov P., p 241 pp 410– (2001)
[14] Koshlukov P., T 128 pp 157– (2002) · Zbl 1049.16014
[15] Krakowski D., Trans. Am. Math. Soc. 181 pp 429– (1973)
[16] DOI: 10.1007/BF01980637 · Zbl 0521.16014 · doi:10.1007/BF01980637
[17] DOI: 10.1007/BF02218641 · Zbl 0282.17003 · doi:10.1007/BF02218641
[18] DOI: 10.1016/0021-8693(90)90286-W · Zbl 0738.16007 · doi:10.1016/0021-8693(90)90286-W
[19] DOI: 10.1080/00927879108824231 · Zbl 0733.16008 · doi:10.1080/00927879108824231
[20] Stojanova-Venkova A. H., Serdica 6 pp 63– (1980)
[21] Vasilovsky S. Yu., Commun. Alg. 26 pp 601– (1998)
[22] DOI: 10.1090/S0002-9939-99-04986-2 · Zbl 0935.16012 · doi:10.1090/S0002-9939-99-04986-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.