Abstract Algebra

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 […]

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 […]

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 […]

Here we showed that if is a PI ring satisfying a polynomial of degree then also satisfies a monic multilinear polynomial There is one particular multilinear polynomial that we’re interested in; it’s called the standard identity. Definition. The standard identity is a multilinear polynomial of degree defined as follows where is the symmetric group of […]

Here we defined the Dirichlet ring and showed that is a local ring. In this post, we show that is the ring of formal power series in infinitely many variables over Notation. In the proof of the following Proposition, denote, respectively, the -th prime number and the exponent of in the prime factorization of an […]