Ring Extension Problem, Shukla Cohomology and Ann-category Theory
نویسندگان
چکیده
Every ring extension of A by R induces a pair of group homomorphisms L : R → EndZ(A)/L(A);R : R → EndZ(A)/R(A), preserving multiplication, satisfying some certain conditions. A such 4-tuple (R,A,L,R) is called a ring pre-extension. Each ring pre-extension induces a R-bimodule structure on bicenter KA of ring A, and induces an obstruction k, which is a 3-cocycle of Z-algebra R, with coefficients in R-bimodule KA in the sense of Shukla. Each obstruction k in this sense induces a structure of a regular Ann-category of type (R,KA). This result gives us the first application of Anncategory in extension problems of algebraic structures, as well as in cohomology theories.
منابع مشابه
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An Ann-category is a categorification of rings. Regular Ann-categories were classified by Shukla cohomology of algebras. In this paper, we state and prove the precise theorem on classification of Ann-categories in the general case based on Mac Lane cohomology of rings. AMS subject classifications: 18D10, 16E40
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