Abstract Algebra Blog
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Blog covers mathematical topics including basic Algebra, number theory, ring theory and more. Abstract Algebra is a blog by Yaghoub Sharifi, a Iranian-Canadian with PhD in mathematics from Simon Fraser University, Canada.
Abstract Algebra Blog
6d ago
As we defined here, given an integer we say that a group is -abelian if for all Here we showed that if where is the center of is abelian and if is odd, then is -abelian. We now prove a much more interesting result. Proposition. Let be a group with the center If then is ..read more
Abstract Algebra Blog
1w ago
For a division ring we denote by and the center and the multiplicative group of respectively. Let be a division ring, and suppose that is a proper subdivision ring of i.e. is a subring of and itself is a division ring. Then is clearly a proper subgroup of Now, one may ask: when exactly is ..read more
Abstract Algebra Blog
1w ago
If is a square matrix with entries from a field of characteristic zero such that the trace of is zero for all positive integers then is nilpotent. This is a very well-known result in linear algebra and there are at least two well-known proofs of that (see here for example) that use the Vandermonde determinant ..read more
Abstract Algebra Blog
3w ago
Throughout this post, is a field and the multiplicative group of We have already seen the general linear group and the special linear group in this blog several times. The general linear group is the (multiplicative) group of all invertible matrices with entries from and the special linear group is the subgroup of consisting of ..read more
Abstract Algebra Blog
3w ago
If is a square matrix with entries from a field of characteristic zero such that the trace of is zero for all positive integers then is nilpotent. This is a very well-known result in linear algebra and there are at least two well-known proofs of that (see here for example) that use the Vandermonde determinant ..read more
Abstract Algebra Blog
1M ago
We remarked here that the composition of derivations of a ring need not be a derivation. In this post, we prove this simple yet interesting result that if is a prime ring of characteristic and if are nonzero derivations of then can never be a derivation of But before getting into the proof of that ..read more
Abstract Algebra Blog
1M ago
All rings in this post are commutative with identity. For the basics on derivations of rings see this post and this post. Let be a derivation of a ring Since is additive, for all integers and all So every derivation of a ring is a -derivation. If is -algebra and where are integers and then ..read more
Abstract Algebra Blog
1M ago
See the first part of this post here. All rings in this post are assumed to have identity. In the first part, we gave two examples of derivations on rings. We now give a couple of ways to make new derivations using given ones. Example 1. Let be a ring, and let the center of ..read more
Abstract Algebra Blog
1M ago
All rings in this post are assumed to have identity and denotes the center of a ring Those who have the patience to follow my posts have seen derivation on rings several times but they have not seen a post exclusively about the basics on derivations of rings because, strangely, that post didn’t exist until ..read more
Abstract Algebra Blog
1M ago
Let be the ring of matrices with real entries. A form of the following problem was posted on the Art of Problem Solving website a couple of days ago. Problem. Show that if and then Solution (Y. Sharifi). Let where Let Then Thus, since and we get that Therefore which gives Now, if then and ..read more