Rings and algebras

the set $\mathbb {Z}$;

the set $n \mathbb {Z}(n>1)$;

the set of nonnegative integers;

the set $\mathbb {Q}$;

the set of rational numbers whose denominators, in lowest terms, divide a fixed number $n \in \mathbb {N}$;

the set of rational numbers whose denominators, in lowest terms, are not divisible by a fixed prime number $p$;

the set of rational numbers whose denominators, in lowest terms, are powers of a fixed prime number $p$;

the set of real numbers of the form $x+y \sqrt{2}$, where $x, y \in \mathbb {Q}$;

the set of real numbers of the form $x+y \sqrt[3]{2}$, where $x, y \in \mathbb {Q}$;

the set of real numbers of the form $x+y \sqrt[3]{2}+z \sqrt[3]{4}$, where $x, y$, $z \in \mathbb {Q}$;

the set of complex numbers of the form $x+y i$, where $x, y \in \mathbb {Z}$;

the set of complex numbers of the form $x+y i$, where $x, y \in \mathbb {Q}$;

the set of all sums of the form $a_{1} z_{1}+a_{2} z_{2}+\ldots +a_{n} z_{n}$, where $a_{1}, a_{2}, \ldots , a_{n}$ are rational numbers and $z_{1}, z_{2}, \ldots , z_{n}$ are the complex $n$-th roots of 1 ;

the set of complex numbers of the form $\frac{x+y \sqrt{D}}{2}$, where $D$ is a fixed square-free integer (not divisible by the square of a prime number), and $x, y$ are integers of the same parity?

the set of real symmetric matrices of order $n$;

the set of real orthogonal matrices of order $n$;

the set of upper triangular matrices of order $n \geqslant 2$;

the set of matrices of order $n \geqslant 2$ whose last two rows are zero;

the set of matrices of the form $\left(\begin{array}{cc}x & y \\ D y & x\end{array}\right)$, where $D$ is a fixed integer, $x, y \in \mathbb {Z}$;

the set of matrices of the form $\left(\begin{array}{cc}x & y \\ D y & x\end{array}\right)$, where $D$ is a fixed element of some ring $K, x, y \in K$;

the set of matrices of the form $\frac{1}{2}\left(\begin{array}{cc}x & y \\ D y & x\end{array}\right)$, where $D$ is a fixed square-free integer, and $x$ and $y$ are integers of the same parity;

the set of complex matrices of the form $\left(\begin{array}{cc}z & w \\ -\bar{w} & \bar{z}\end{array}\right)$;

the set of real matrices of the form

the set of functions of a real variable, continuous on the interval $[a, b]$;

the set of functions having a second derivative on the interval $(a, b)$;

the set of polynomial functions of a real variable;

the set of rational functions of a real variable;

the set of functions of a real variable vanishing on some subset $D \subseteq \mathbb {R}$;

the set of trigonometric polynomials

with real coefficients, where $n$ is an arbitrary natural number;

the set of trigonometric polynomials of the form

with real coefficients, where $n$ is an arbitrary natural number;

the set of trigonometric polynomials of the form

with real coefficients, where $n$ is an arbitrary natural number;

the set of functions defined on some set $D$ and taking values in some ring $R$;

all formal power series in one or several variables;

all Laurent power series in one variable?

63.1, (n) and 63.2, (g);

63.2, (h) and 63.2, (i).

$\mathbf{Z}_{n}$;

$\mathbf{Z}_{p^{n}}$, where $p$ is a prime number;

$K[x] /(f K[x])$, where $K$ is a field;

upper triangular matrices over a field;

$\mathbf{M}_{2}(\mathbb {R})$;

all functions defined on some set $S$ and taking values in a field $K$;

all power series in one variable;

$\mathbb {Z}$;

$\mathbb {Z}\left[i\right]$.

if $R$ has no zero divisors, then it has a unity and all its nonzero elements are invertible;

if $R$ has a unity, then every element of it having a one-sided inverse is invertible;

if $R$ has a unity, then every left zero divisor is a right zero divisor.

if the products $x y$ and $y x$ are invertible, then the elements $x$ and $y$ are also invertible;

if $R$ has no zero divisors and the product $x y$ is invertible, then $x$ and $y$ are invertible;

without additional assumptions on the ring $R$, invertibility of the product $x y$ does not imply invertibility of the elements $x$ and $y$;

if the element $1+a b$ is invertible, then so is the element $1+b a$.

Under what conditions is $R$ commutative; does it have a unity; does it have no zero divisors?

Find all invertible elements of $R$; all zero divisors; all nilpotent elements.

if the numbers $k$ and $l$ are coprime, then $\mathbf{Z}_{k l}=\mathbf{Z}_{k} \oplus \mathbf{Z}_{l}$;

if $n=p_{1}^{k_{1}} \ldots p_{s}^{k_{s}}$, where $p_{1}, \ldots , p_{s}$ are distinct prime numbers, then

if the numbers $k$ and $l$ are coprime, then $\varphi (k l)=\varphi (k) \varphi (l)$, where $\varphi$ is Euler's function.

a zero divisor in an arbitrary (associative) algebra is not invertible;

in a finite-dimensional algebra with unity, every element that is not a zero divisor is invertible;

a finite-dimensional algebra without zero divisors is a division ring (a division algebra).

a finite-dimensional algebra with unity and without zero divisors over the field $\mathbb {C}$ is isomorphic to $\mathbb {C}$;

over the field $\mathbb {C}$ there is no finite-dimensional division algebra other than $\mathbb {C}$.

with unity;

not necessarily with unity.

with unity;

not necessarily with unity.

Is $\mathbb {H}$ an algebra over the field $\mathbb {C}$, if scalar multiplication by $\alpha \in \mathbb {C}$ is understood as left multiplication by $\alpha \in \mathbb {H}$?

Prove that the maps

give an isomorphism of $\mathbb {H}$, as an algebra over the field $\mathbb {R}$, onto some subalgebra of the matrix algebra $\mathbf{M}_{2}(\mathbb {C})$ over $\mathbb {R}$.

Prove that the map $z \mapsto \left(\begin{array}{cc}z & 0 \\ 0 & \bar{z}\end{array}\right)$ is an isomorphic embedding of the field $\mathbb {C}$ into the algebra $\mathbb {H}$, realized as a subalgebra of the algebra $\mathrm{M}_{2}(\mathbb {C})$ over $\mathbb {R}$ (see (b)).

Solve the equation $x^{2}=-1$ in $\mathbb {H}$.

$\mathbb {T}(V)$ is an associative algebra with unity over the field $K$;

$\mathbb {T}(V)$ has no zero divisors.

$\Lambda (V)$ is an associative algebra with unity over the field $K$;

every element of $I=\bigoplus_{k \geqslant 1} \Lambda^{k}(V)$ is nilpotent;

every element of $\Lambda (V)$ not lying in $I$ is invertible.

$S(V)$ is an associative, commutative algebra over $K$;

if $x_{1}, \ldots , x_{n}$ is a basis of the space $V$, then $S(V)$ is isomorphic to the algebra of polynomials in $x_{1}, \ldots , x_{n}$.

$\mathbb {C} \otimes_{K} \mathbb {C} \simeq \mathbb {C} \oplus \mathbb {C} \quad (K=\mathbb {R})$;

$\mathbf{M}_{n}(K) \otimes_{K} \mathbf{M}_{m}(K) \simeq \mathbf{M}_{m n}(K)$;

$\mathbf{M}_{n}(K) \otimes_{K} A \simeq \mathbf{M}_{n}(A)$, where $A$ is an arbitrary associative algebra over $K$;

$K\left[X_{1}, \ldots , X_{n}\right] \otimes_{K} K\left[Y_{1}, \ldots , Y_{m}\right] \simeq K\left[X_{1}, \ldots , X_{n}, Y_{1}, \ldots , Y_{m}\right] ;$

$\mathbb {H} \otimes_{\mathbb {R}} \mathbb {C} \simeq \mathbf{M}_{2}(\mathbb {C})$;

$\mathbf{S}(V) \otimes_{K} \Lambda (V) \simeq \mathbb {T}(V)$ for $\operatorname {dim} V=2$;

$\mathbb {Q}(\sqrt{p}) \otimes \mathbb {Q}(\sqrt{q}) \simeq \mathbb {Q}(\sqrt{p}+\sqrt{q})$, where $p$ and $q$ are distinct prime numbers.

$q_{j} p_{i}-p_{i} q_{j}=\delta_{i j}, \quad p_{i} p_{j}=p_{j} p_{i}, \quad q_{i} q_{j}=q_{j} q_{i}$;

a basis of $A_{n}(K)$ as a vector space is formed by the monomials

in the ring of all functions on the interval $[0,1]$, the zero divisors are exactly the functions taking the value zero somewhere;

in the ring of continuous functions on the interval $[0,1]$, the zero divisors are the nonzero functions that vanish on some interval $[a, b]$, where $0 \leqslant a<b \leqslant 1$.