Articles of abstract algebra

Irreducible Components of the Prime Spectrum of a Quotient Ring and Primary Decomposition

Recently I encountered a problem (the first exercise from chapter four of Atiyah & McDonald’s Introduction to Commutative Algebra) stating that if $\mathfrak{a}$ is a decomposable ideal of $A$ (a commutative ring with unity), then the prime spectrum of $A/ \mathfrak{a}$ has finitely many irreducible components. This follows easily from the recognition that the maximal […]

Kaplansky's theorem of infinitely many right inverses in monoids?

There’s a theorem of Kaplansky that states that if an element $u$ of a ring has more than one right inverse, then it in fact has infinitely many. I could prove this by assuming $v$ is a right inverse, and then showing that the elements $v+(1-vu)u^n$ are right inverses for all $n$ and distinct. To […]

Centre of a matrix ring are diagonal matrices

This question already has an answer here: A linear operator commuting with all such operators is a scalar multiple of the identity. 9 answers

What is an Homomorphism/Isomorphism “Saying”?

Outside of the technical definitions, what exactly is a homormorphism or an isomorphism “saying”? For instance, let’s we have a group or ring homomorphism $f$, from $A$ to $B$. Does a homomorphism mean that $f$ can send some $a_i$ in $A$ to $b_j$ in $B$, but has no way to “get it back”? Similarly, if […]

Constructing Isomorphism between finite field

Consider $\mathbb{F}_3(\alpha)$ where $\alpha^3 – \alpha +1 = 0$ and $\mathbb{F}_3(\beta)$ where $\beta^3 – \beta^2 +1 =0$. I know these two fields are isomorphic but I have difficulty buliding an isomorphism between them. I know I have to determine where $\alpha$ is mapped to under the isomorphism map but I can’t figure it out. Any […]

If I know the order of every element in a group, do I know the group?

Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ have the same number of elements for each order, but $G$ is not isomorphic to […]

If $P \leq G$, $Q\leq G$, are $P\cap Q$ and $P\cup Q$ subgroups of $G$?

$P$ and $Q$ are subgroups of a group $G$. How can we prove that $P\cap Q$ is a subgroup of $G$? Is $P \cup Q$ a subgroup of $G$? Reference: Fraleigh p. 59 Question 5.54 in A First Course in Abstract Algebra.

Abelian $p$-group with unique subgroup of index $p$

Let $G$ be a finite abelian $p$-group with a unique subgroup $H$ of index $p$. It is a fact that $G$ is cyclic. This can be deduced from the classification theorem for finite abelian groups by writing $G$ as a product of cyclic groups and noting that $G$ does not have a unique subgroup of […]

Norm-Euclidean rings?

For which integer $d$ is the ring $\mathbb{Z}[\sqrt{d}]$ norm-Euclidean? Here I’m referring to $\mathbb{Z}[\sqrt{d}] = \{a + b\sqrt{d} : a,b \in \mathbb{Z}\}$, not the ring of integers of $\mathbb{Q}[\sqrt{d}]$. For $d < 0$, it is easy to show that only $d = -1, -2$ suffice; but what about $d>0$? Thanks.

How many idempotent elements does the ring ${\bf Z}_n$ contain?

Let $R$ be a ring. An element $x$ in $R$ is said to be idempotent if $x^2=x$. For a specific $n\in{\bf Z}_+$ which is not very large, say, $n=20$, one can calculate one by one to find that there are four idempotent elements: $x=0,1,5,16$. So here is my question: Is there a general result which […]