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Title: Abstract Algebra

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PI rings; the standard identity (2)
Here we defined the standard identity and showed that is alternating, i.e., if for some then In this post, we look at relationships between and and we show that if a ring satisfies for some then it will satisfy for all Proposition. For any positive integer Proof. By definition, Now, for a fixed positive integer … …

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A remark on centrally module-finite rings
Let be a ring with identity with the center Let be the ring of -module homomorphisms It is a well-known fact that is embedded in we’re going to prove this shortly. We are also going to show that can not always be embedded in a matrix ring over even if we assume that as a […]

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Which commutative domains satisfy some nonzero polynomial?
Let be a commutative domain with identity, and let be some commuting indeterminates over Let be the ring of polynomials with coefficients in We say that satisfies if for all Problem. Show that satisfies some nonzero polynomial in if and only if is a finite field. Solution. Recall that a finite commutative domain is a […]

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Ring of endomorphisms of module-finite rings over commutative subrings
Throughout this note, is a ring with identity, is a commutative subring of and as a left -module, is finitely generated. As usual, denote, respectively, the ring of -module homomorphisms and the ring of matrices with entries from The goal in this post is to find in terms of First note that if is free […]

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Matrix rings over commutative rings are PI
In this note, is a commutative ring with identity, and is the ring of matrices with entries from Also, is the standard identity of degree Here we defined PI rings and gave some examples including We now show that is PI for all Proposition. Let be a ring, and let be a central subring of […]

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