Ideals

Let $A$ be the ring of real $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ 0 & c \end{pmatrix}$. What are the 2-sided ideals in $A$?

Let $M_{n \times n}(\mathbf{F})$ be the ring of $n \times n$ matrices over a field $\mathbf{F}$. Prove that it has no 2-sided ideals except $M_{n \times n}(\mathbf{F})$ and $\left\{ 0\right\}$.

Let $M_{n \times n}(\mathbf{F})$ be the ring of $n \times n$ matrices over a field $\mathbf{F}$. For $n \geq 1$ does there exist a ring homomorphism from $M_{(n+1) \times (n+1)}(\mathbf{F})$ onto $M_{n \times n}(\mathbf{F})$?

Let $R$ be the ring of $n \times n$ matrices over a field. Suppose $S$ is a ring and $h : R \rightarrow S$ is a homomorphism. Show that $h$ is either injective or zero.

Let $\mathbf{F}$ be a field and let $X$ be a finite set. Let $R(X, \mathbf{F})$ be the ring of all functions from $X$ to $\mathbf{F}$, endowed with the pointwise operations. What are the maximal ideals of $R(X, \mathbf{F})$?

Suppose $V$ is a vector space over a field $K$. If $U$ and $W$ are subspaces, let $E(U,W)$ be the set of linear endomorphisms $F$ of $V$ over $K$ with the property that the image of $FU$ in $V/W$ is finite dimensional. Show that $E(U,U)$ is a subring of the ring of endomorphisms of $V$ with two-sided ideals $E(V,U)$ and $E(U,0)$.

Let $R$ be a commutative ring with identity element and $a \in R$. Let $n$ and $m$ be positive integers, and write $d = \gcd \left\{ n,m\right\}$. Prove that the ideal of $R$ generated by $a^n - 1$ and $a^m - 1$ is the same as the ideal generated by $a^d - 1$.

Let $R$ be a commutative ring with 1, and $R^{*}$ be its group of units. Suppose that the additive group of $R$ is generated by $\left\{ u^2 \mid u \in R^{*}\right\}$. Prove that $R$ has, at most, one ideal $I$ for which $R/I$ has cardinality 3.

Find all commutative rings $R$ with identity such that $R$ has a unique maximal ideal and such that the group of units of $R$ is trivial.

Let $R$ be a ring with identity, and let $I$ be the left ideal of $R$ generated by $\left\{ ab-ba \mid a,b \in R\right\}$. Prove that $I$ is a two-sided ideal.

Let $R$ be a ring with identity element. Suppose that $I_1, I_2, \ldots , I_n$ are left ideals in $R$ such that $R = I_1 \oplus I_2 \oplus \cdots \oplus I_n$ (as additive groups). Prove that there are elements $u_i \in I_i$ such that for any elements $a_i \in I_i$, $a_i u_i = a_i$ and $a_i u_j = 0$ if $j \neq i$.

Suppose that $R$ is a subring of a commutative ring $S$ and that $R$ is of finite index $n$ in $S$. Let $m$ be an integer that is relatively prime to $n$. Prove that the natural map $R/mR \rightarrow S/mS$ is a ring isomorphism.

Let $M$ be one of the following fields: $\mathbb {R}$, $\mathbb {C}$, $\mathbb {Q}$, and $\mathbf{F}_9$ (the field with nine elements). Let $I \subset M[x]$ be the ideal generated by $x^4 + 2x - 2$. For which choices of $M$ is the ring $M[x]/I$ a field?

Let $R$ be a principal ideal domain and let $I$ and $J$ be nonzero ideals in $R$. Show that $IJ = I \cap J$ if and only if $I+J = R$.