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Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
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A finite direct product of Azumaya algebras may not be Azumaya?
Looking for an example, if it exists, of a finite set of Azumaya algebras $A_i$ such that $\oplus A_i$ is not Azumaya.
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localization at central maximal ideals
If M is a maximal ideal of Z(R), the center of a ring with identity R, and R_M, the localization of R at M, is a commutative field, what can we say about R?
My guess is that we can's say that much be …