Abstract Algebra

For a ring and a set we denote by the center of i.e., It is clear that for any In this post, we take a quick look into the class of rings with this property that for every left or right ideal of Let’s begin with labeling such rings. Notation. Throughout this note, denotes the […]

Proposition. Let be a domain. If is PI, then is Ore. Proof. We give two proofs of the Proposition. First Proof. Let be the center of and suppose to the contrary that is not Ore. Then, by the Lemma in this post, contains a copy of the noncommutative polynomial ring So being a subring of […]

As always we denote by the ring of matrices with entries from a ring Let be a commutative ring with identity, and let be a -algebra. Here we defined PI rings and PI -algebras; let me repeat that again. We say that is a PI ring if it satisfies some monic polynomial with coefficients in […]

Here we showed that every PI ring satisfies some monic multilinear polynomial. The standard identity is the most important monic multilinear polynomial in the theory of PI rings. Now, you may ask: does every PI ring satisfy some standard identity? The answer is negative. In this note, we are going to see an example of […]

Let be the symmetric group of degree It is clear that every element of is a product of disjoint cycles. So, since every cycle is a product of transpositions, because every element of is a product of transpositions. An adjacent transposition is any transposition of the form Every transposition is a product of adjacent transpositions […]