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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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Morphisms in K-theory: comparison of two pictures
It seems that by $M_n(A)$ you denote two different objects: actual $M_n(A)$ -- which is the same as $\text{Hom}(A^n, A^n)$ -- and $A^n$ itself.
One can show that any finitely generated projective $A$ …