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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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Matrix Algebras
I'm reading the papaer "On the Reduction of a Matrix to Diagonal Form" of Epstein and Flanders (Amer. Math. Monthly 62, (1955). 168–171.
Let $S$ denote the trace function.
The authors stated that a …
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Commutative Subrings of Finite Matrix Rings
Let $R$ be a finite commutative ring with identity. Considere the matrix ring $A=M_n(R)$. What is the order of a maximal commutative subring of $A$ that contains all scalar matrices?
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Annihilators in Matrix Rings
I think this is not a research question, but in stackExchange remained unanswered.
Let $R$ be a finite commutative ring. For $n>1$ consider the full matrix ring $M_n(R)$ . For a matrix $A\in M_n(R) …
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Matrices over Finite Prime Fields
Let $p$ be an odd prime and $\mathbb Z_p$ be the prime field of order $p$. Consider the matrix ring $R=M_n(\mathbb Z_p)$. Is there any method to count the solutions of the equation (in the ring $R$)
…
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Embedding Semigroups in Rings
Let $S$ be a finite commutative semigroup with identity. Under what conditions (on the semigroup $S$) it is possible to find a ring $R$ such that the multiplicative structure of $R - \{0\}$ is isomorp …
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Centralizer of a Matrix over a Finite Field
This question in stackExchange remained unanswered.
Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$ . For a matrix $A∈M_n(\mathbb F …