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Algebras that are sums of two PI subalgebras

Grant number: 19/16994-1
Support type:Scholarships in Brazil - Doctorate
Effective date (Start): October 01, 2019
Effective date (End): February 28, 2023
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Algebra
Principal Investigator:Plamen Emilov Kochloukov
Grantee:Pedro Souza Fagundes
Home Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
Associated research grant:18/23690-6 - Structures, representations, and applications of algebraic systems, AP.TEM


In this research proposal we plan to study the following problem. Let R be an associative ring suc that RR_1R_2 where R_1 and R_2 are subrings of R. The sum need not be direct nor R_1 and/or R_2 need be ideals in R. Suppose tat R_1 and R_2 both satisfy polynomial identities (PI), can one conclude R is a PI ring or not. This problem is quite old, there is an extensive list of papers dealing with it but imposing some restrictions on R_1 and/or R_2. Recently (in 2017), Marek K\c{e}pczyk obtained the positive answer in the general situation for associative rings. His paper albeit quite short, was based heavily on several other papers. Thus the length of that paper cannot give a precise measure of the depth and of the importance of his result. The theorem of K\c{e}pczyk makes heavy use of the structure theory of associative rings and algebras. Our goals will comprise the study of associative rings graded by a finite group G, and establish whether an analogue of the above mentioned result holds in this case. Recall that the structure theory of graded associative rings and algebras is well developed and parallels that in the ungraded case. We expect that if G is finite and the subrings of R are homogeneous in the grading the answer will be also positive, and R will satisfy graded identities. Afterwards we will be studying the same problem for the case of Lie algebras. Recall that in this case the existence of a PI is not such a strong restriction as it is in the associative case. We suspect that in the Lie case the answer to the main problem will be negative. The same problem can be posed for restricted (p-) Lie algebras, and also for graded Lie algebras and for Lie superalgebras. We expect that in the restricted case the answer may be positive while in the superalgebra case the answer might be negative. We stress that these last cases will be studied if the first parts of the project run as we expect. (AU)