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|G/Z(G)| = n implies G is n-abelian | Abstract Algebra
As we defined here, given an integer $latex n,$ we say that a group $latex G$ is $latex n$-abelian if $latex (xy)^n=x^ny^n$ for all $latex x,y \in G.$ Here we showed that if $latex G/Z(G),$ where $latex Z(G)$ is the center of $latex G,$ is abelian and if $latex |Z(G)|=n$ is odd, then $latex G$…
A simple property of Euclidean domains | Abstract Algebra
As always, given a ring $latex R,$ we denote by $latex U(R)$ the multiplicative group of units of $latex R.$ Definition. Let $latex R$ be a commutative domain with identity. An element $latex a \in R \setminus U(R)$ is called a universal side divisor if for every $latex b \in R,$ either $latex a \mid…
|G/Z(G)| = n ⇒ G is n-abelian | Abstract Algebra
As we defined here, given an integer $latex n,$ we say that a group $latex G$ is $latex n$-abelian if $latex (xy)^n=x^ny^n$ for all $latex x,y \in G.$ Here we showed that if $latex G/Z(G),$ where $latex Z(G)$ is the center of $latex G,$ is abelian and if $latex |Z(G)|=n$ is odd, then $latex G$…
If |G/Z(G)| = n, then G is n-abelian | Abstract Algebra
As we defined here, given an integer $latex n,$ we say that a group $latex G$ is $latex n$-abelian if $latex (xy)^n=x^ny^n$ for all $latex x,y \in G.$ Here we showed that if $latex G/Z(G),$ where $latex Z(G)$ is the center of $latex G,$ is abelian and if $latex |Z(G)|=n$ is odd, then $latex G$…
The Cartan-Brauer-Hua theorem | Abstract Algebra
For a division ring $latex D,$ we denote by $latex Z(D)$ and $latex D^{\times}$ the center and the multiplicative group of $latex D,$ respectively. Let $latex D_1$ be a division ring, and suppose that $latex D_2 $ is a proper subdivision ring of $latex D_1,$ i.e. $latex D_2$ is a subring of $latex D_1, \…
A matrix is nilpotent iff the trace of every power of the matrix is zero | Abstract Algebra
If $latex A$ is a square matrix with entries from a field of characteristic zero such that the trace of $latex A^m$ is zero for all positive integers $latex m,$ then $latex A$ is nilpotent. This is a very well-known result in linear algebra and there are at least two well-known proofs of that (see…
Group homomorphisms from GL(n,k) to the multiplicative group of k | Abstract Algebra
Throughout this post, $latex k$ is a field and $latex k^{\times}:=k \setminus \{0\},$ the multiplicative group of $latex k.$ We have already seen the general linear group $latex \text{GL}(n,k)$ and the special linear group $latex \text{SL}(n,k)$ in this blog several times. The general linear group $latex \text{GL}(n,k)$ is the (multiplicative) group of all $latex n…
A matrix is nilpotent iff the trace of every positive power of the matrix is zero | Abstract Algebra
If $latex A$ is a square matrix with entries from a field of characteristic zero such that the trace of $latex A^m$ is zero for all positive integers $latex m,$ then $latex A$ is nilpotent. This is a very well-known result in linear algebra and there are at least two well-known proofs of that (see…
Composition of derivations in prime rings; a theorem of Posner | Abstract Algebra
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 $latex R$ is a prime ring of characteristic $latex \ne 2$ and if $latex \delta_1,\delta_2$ are nonzero derivations of $latex R,$ then $latex \delta_1\delta_2$ can never be…
Derivations on rings; the basics (1) | Abstract Algebra
All rings in this post are assumed to have identity and $latex Z(R)$ denotes the center of a ring $latex R.$ 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…
Derivations on rings; the basics (2) | Abstract Algebra
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 $latex R$ be a ring, and let $latex…
ℚ-algebras with locally nilpotent derivations | Abstract Algebra
All rings in this post are commutative with identity. For the basics on derivations of rings see this post and this post. Let $latex \delta$ be a derivation of a ring $latex R.$ Since $latex \delta$ is additive, $latex \delta(na)=n\delta(a)$ for all integers $latex n$ and all $latex a \in R.$ So every derivation of…
If real matrices A, B satisfy A^2 + B^2 = AB, then det(BA – AB) ≥ 0 | Abstract Algebra
Let $latex M_n(\mathbb{R})$ be the ring of $latex n \times n$ 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 $latex A,B \in M_n(\mathbb{R})$ and $latex A^2+B^2=AB,$ then $latex \det(BA-AB) \ge 0.$ Solution (Y. Sharifi). Let…
PI rings; definition and basic examples | Abstract Algebra
Rings in this post may or may not have identity. As always, the ring of $latex n \times n$ matrices with entries from a ring $latex R$ is denoted by $latex M_n(R).$ PI rings are arguably the most important generalization of commutative rings. This post is the first part of a series of posts about…
Permutation matrix; an alternative view | Abstract Algebra
As always, in this post, we denote by $latex \text{GL}(n,k)$ the group of $latex n \times n$ invertible matrices with entries from a field $latex k.$ Also, $latex A^T$ is the transpose of a matrix $latex A.$ Here we defined a permutation matrix as an $latex n \times n$ matrix $latex A$ such that $latex…
Left-multiplication maps as ring homomorphisms | Abstract Algebra
Throughout this post, $latex R$ is a ring with identity and $latex Z(R)$ is its center. We denote by $latex \mathcal{I}(R)$ the set of idempotents of $latex R,$ i.e. $latex r \in R$ such that $latex r^2=r.$ Also, $latex R[x]$ is the ring of polynomials in $latex x$ with coefficients in $latex R.$ Given $latex…
Semiprime rings | Abstract Algebra
Throughout this post, $latex R$ is a ring with identity, and $latex M_n(R)$ is the ring of $latex n \times n$ matrices with entries from $latex R.$ Also, by ideal we always mean two-sided ideal. Here we defined a prime ideal of $latex R$ as a proper ideal $latex P$ which satisfies this condition: if…
Prime rings | Abstract Algebra
Throughout this post, $latex R$ is a ring with identity, and $latex M_n(R)$ is the ring of $latex n \times n$ matrices with entries from $latex R.$ Also, by ideal we always mean two-sided ideal. In commutative ring theory, a proper ideal $latex P$ of a commutative ring $latex R$ is said to be prime…
Every nil, multiplicatively closed set of matrices is nilpotent | Abstract Algebra
Throughout this post, $latex k$ is a field and $latex M_n(k)$ is the ring of $latex n \times n$ matrices with entries in $latex k.$ We have already seen the concepts of nil and nilpotency in a ring in this blog several times, but let's see it here again. Let $latex R$ be a ring,…
Finite sum of reciprocals of squares of integers modulo primes of the form 6k + 1 | Abstract Algebra
Let's begin this post with an extension of the concept of divisibility of integers by a prime number to divisibility of positive rational numbers by a prime number. Definition. Let $latex p$ be a prime number, and $latex r=\frac{a}{b},$ where $latex a,b$ are positive integers and $latex p \nmid b.$ Then we say that $latex…
Rings satisfying (xy)^n=x^ny^n, for all n, may not be commutative | Abstract Algebra
In this post, we took a look at groups $latex G$ satisfying $latex (xy)^n=x^ny^n$ for some integer $latex n$ and all $latex x,y \in G.$ We showed that if $latex n=2,$ then $latex G$ is abelian, but for $latex n > 2,$ the group $latex G$ may or may not be abelian. We can now…
Integro-differential rings | Abstract Algebra
Throughout, $latex R$ is a ring. Consider the ordinary polynomial ring $latex \mathbb{R}[x].$ In this ring, we have the familiar concepts of differentiation and integration. In this post, we define those concepts in arbitrary rings. Let's begin with derivation, which we have already seen in this blog several times. Definition 1. A derivation on $latex…
Ideals of a polynomial ring R[x] which are of the form I[x] for some ideal I of R | Abstract Algebra
Let $latex R$ be a commutative ring with identity, and let $latex I$ be an ideal of $latex R.$ Let $latex R[x]$ be the ring of polynomials over $latex R,$ and let $latex I[x]$ be the set of all polynomials in $latex R[x]$ with all coefficients in $latex I.$ So an element of $latex I[x]$…
Euclidean domains with a special Euclidean function | Abstract Algebra
In this post we are going to show that if $latex R$ is a Euclidean domain with a Euclidean function $latex \phi$ that satisfies the inequality $latex \phi(a+b) \le \max \{\phi(a),\phi(b)\},$ whenever $latex a,b,a+b \in R \setminus \{0\},$ then $latex R$ is either a field or a polynomial ring over a field. This result was…
Solvable groups; the Burnside’s Problem | Abstract Algebra
Let $latex G$ be a group. Recall that an element of $latex G$ is said to be torsion if it has finite order. We say that $latex G$ is torsion if every element of $latex G$ is torsion. Recall also that $latex G$ is called finitely generated if there exists a finite subset $latex S$…
In a finitely generated group, subgroups of finite index are finitely generated | Abstract Algebra
Let $latex G$ be a group. Recall that $latex G$ is said to be finitely generated if there exists a finite subset $latex S$ of $latex G$ such that every element of $latex G$ is a finite product of some elements of $latex S \cup S^{-1},$ where $latex S^{-1}:=\{s^{-1}: \ s \in S\}.$ In this…
Fermat’s last theorem does not hold for integer matrices | Abstract Algebra
Fermat's last theorem states that if $latex n \ge 3$ is an integer, then there are no positive integers $latex a,b,c$ such that $latex a^n+b^n=c^n.$ This problem remained unsolved for 3 centuries but 3 decades ago, Andrew Wiles solved it. Now, let's replace positive integers with $latex n \times n$ matrices with integer entries and…
The ring of real-valued continuous functions on the interval [0,1] | Abstract Algebra
Let $latex C$ be the set of all continuous functions $latex f:[0,1] \to \mathbb{R}.$ We begin this post with turning $latex C$ into a commutative ring with identity. For any $latex f,g \in C,$ define $latex f+g$ and $latex fg$ by $latex (f+g)(x)=f(x)+g(x), \ \ \ \ \ \ (fg)(x)=f(x)g(x), \ \ \ \ \forall…
Tensor product of finite abelian groups | Abstract Algebra
Here we defined tensor product of modules over commutative rings. Since abelian groups, considered additively, are just $latex \mathbb{Z}$-modules, tensor product of abelian groups is just tensor product of $latex \mathbb{Z}$-modules, which is again a $latex \mathbb{Z}$-module. So the tensor product of two abelian groups is always an abelian group. As we are going to…
Schur’s inequality | Abstract Algebra
As always, $latex M_n(\mathbb{C})$ denotes the ring of $latex n \times n$ matrices with complex entries, and $latex \lfloor \ \rfloor$ is the floor function. Schur was surely a talented mathematician; we have already seen two of his results in this blog: Schur's lemma and Schur's theorem. In this post, we prove Schur's inequality. In…
Which groups are the union of three proper subgroups? | Abstract Algebra
Throughout this post, $latex V$ denotes the Klein four-group. In this post, we showed that no group is the union of two proper subgroups but there are groups that are the union of three proper subgroups, and we gave $latex V$ as an example. The Klein four-group $latex V$ is the non-cyclic group of order…
Groups in which all non-identity elements have the same order | Abstract Algebra
Notation 1. Throughout this post, $latex \mathcal{E}$ denotes the class of all groups with this property that all the non-identity elements of the group have the same order. Here by "order", we mean finite order. Abelian elements of $latex \mathcal{E}$ are called elementary abelian groups. We begin this post by showing that if all the…
Finite covers of groups by cosets; the Neumann’s lemma | Abstract Algebra
Let $latex G$ be a group. If $latex H$ is a subgroup of $latex G$ and $latex G=\bigcup_{i=1}^nHx_i, \ x_i \in G,$ then, by the definition of the index of a subgroup, we have $latex [G:H] \le n.$ Now, consider a general case. Let $latex H_1, H_2, \cdots , H_n$ be not necessarily distinct subgroups…
Finite covers of groups by cosets; Neumann’s lemma | Abstract Algebra
Let $latex G$ be a group. If $latex H$ is a subgroup of $latex G$ and $latex G=\bigcup_{i=1}^nHx_i, \ x_i \in G,$ then, by the definition of the index of a subgroup, we have $latex [G:H] \le n.$ Now, consider a general case. Let $latex H_1, H_2, \cdots , H_n$ be not necessarily distinct subgroups…
Integers not divisible by 4 as sum of 4 squares | Abstract Algebra
Which (positive) integers can be written as sum of $latex 2$ squares? This question was answered by the great Swiss mathematician Leonhard Euler: integers that are of the form $latex km^2$ for some positive integers $latex k,m$ and no prime divisor of $latex k$ is $latex 3 \mod 4.$ This was probably one of the…
Integers not divisible by 4 as sum of four squares | Abstract Algebra
Which (positive) integers can be written as sum of two squares? This question was answered by the great Swiss mathematician Leonhard Euler: integers that are of the form $latex km^2$ for some positive integers $latex k,m$ and no prime divisor of $latex k$ is $latex 3 \mod 4.$ This was probably one of the easiest…
Determinant of block matrices | Abstract Algebra
As always, $latex \text{adj}(A)$ denotes the adjugate of a square matrix $latex A.$ Theorem. Let $latex A,B,C,D$ be, respectively, $latex n \times n, \ n \times m, \ m \times n$ and $latex m \times m$ matrices with entries from $latex \mathbb{C}.$ Consider the following $latex (m+n) \times (m+n)$ matrix $latex X=\begin{pmatrix}A & B \\…
The adjugate of a matrix is a polynomial function of the matrix | Abstract Algebra
If you're not familiar with rings, you may just take $latex R$ in the following Problem to be the set of real or complex numbers. Problem. Let $latex R$ be a commutative ring with identity, and let $latex A$ be an $latex n \times n$ matrix with entries from $latex R.$ Let $latex I$ be…
Intersection of a 2-generated subgroup with the commutator subgroup of the group | Abstract Algebra
Let $latex G$ be a group, and $latex x,y \in G.$ Recall that the commutator subgroup $latex G'$ of $latex G$ is the subgroup generated by the set $latex \{[a,b]: \ a,b \in G\},$ where $latex [a,b]:=aba^{-1}b^{-1}.$ So $latex xyx^{-1}y^{-1} \in G'.$ Now, let's consider the element $latex z:=x^my^nx^py^q,$ where $latex m,n,p,q$ are any integers,…
Back to the basics; the subrings’ identity crisis | Abstract Algebra
Recall the definition of a ring: a set $latex R$ with two binary operations, called addition and multiplication, is a ring if $latex (R,+)$ is an abelian group, and multiplication is both associative and distributive over addition. We do not require $latex R$ to have a multiplicative identity, which is denoted by $latex 1$ or…