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\listok{9}{Algebra 9: Artinian rings and idempotents}

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\begin{opredelenie}
  Consider a commutative algebra $R$ with unity over a field $k$. One
  says that $R$ is a {\bf finitely generated Artinian ring over the
    field $k$} if $R$ is finite-dimensional as a vector space.
\end{opredelenie}

\begin{zadacha}
Consider a linear operator  $A\in \End V$.
Consider a subalgebra of $\End V$ generated by
$k$ and $A$. Prove that this in an Artinian ring over $k$.
\end{zadacha}

\begin{opredelenie}
An element  $r\in R$ of an algebra (or ring) $R$ is called
{\bf nilpotent} if $r^k=0$ for some $k\in \N$.
\end{opredelenie}

\begin{zadacha}
Let $r, r'$ be nilpotent elements in an Artinian ring over a
field. Prove that any linear combination  $r, r'$ is nilpotent. 
\end{zadacha}

\begin{zadacha}
Let $r, r'$ be nilpotent elements in the algebra  $\Mat(V)$. Is $r+r'$
always nilpotent?
\end{zadacha}

\begin{zamechanie}
A nilpotent element in the matrix algebra is called a
{\bf nilpotent operator}.
\end{zamechanie}

\begin{zadacha} 
Let $A\in \End V$ be a nilpotent operator. Prove that there is a chain
of subspaces $V\supset V_1\supset V_2 \supset \dots \supset V_k =0$ in
$V$ such that $A(V_i) = V_{i+1}$.
\end{zadacha}

\begin{zadacha}[!]
Consider a nilpotent operator
$A\in \End V$. Prove that in some basis $A$ has the form:
$$
\begin{pmatrix}
0&*&* &\hdotsfor{1} &*&*&*\\
0&0&* &\hdotsfor{1} &*&*&*\\
0&0&0 &\hdotsfor{1} &*&*&*\\
\vdots&\vdots&\vdots&
\ddots
&\vdots&\vdots&\vdots\\
0&0&0 &\hdotsfor{1} &0&*&*\\
0&0&0 &\hdotsfor{1} &0&0&*\\
0&0&0 &\hdotsfor{1} &0&0&0
\end{pmatrix}
$$
(that is, an upper-triangular matrix with 0 on the diagonal).
Prove that any matrix of this form is nilpotent.
\end{zadacha}

\begin{ukazanie}
Use the previous problem.
\end{ukazanie}

\begin{zadacha}[!]\label{tr=0}
Let $A \in \End V$ be nilpotent operator. Prove that
$\tr(A)=\det(A)=0$ and $\chpoly_A(t)=t^{\dim V}$.
\end{zadacha}

\begin{opredelenie}
Let $R$ be a ring. A subset ${\mathfrak m} \subset R$
is called an {\bf ideal} if the following it has the following
properties: 
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item ${\mathfrak m}$ is closed under addition (that is,
the sum of two elements from ${\mathfrak m}$ belongs to ${\mathfrak
  m}$) 

\item For any $m\in{\mathfrak m}$, $a\in R$ the product
$am$ belongs to $R$.
\end{enumerate}
\end{opredelenie}

\begin{zadacha} Consider a homomorphism of rings $R\arrow R'$.
Prove that the kernel of this homomorphism is an ideal.
\end{zadacha}

\begin{zadacha}\label{field.noideals}
Consider a surjective homomorphism $f:R_1 \to R_2$ of algebras over a field
$k$ and let  $R_1$ be a field. Prove that either $R_2=0$, or $f$ is an
isomorphism. 
\end{zadacha}

\begin{zadacha}
Consider an ideal ${\mathfrak m} \subset R$.
Consider the factor $R/{\mathfrak m}$, that is the set of cosets
of the form $r + {\mathfrak m}$.
Define on $R/{\mathfrak m}$ the natural ring structure.
\end{zadacha}

\begin{opredelenie}
  A ring $R/{\mathfrak m}$ is called a {\bf quotient ring (or factor
    ring)} of the ring $R$.  An ideal is called {\bf prime}, if the
  corresponding quotient ring is non-zero and has no zero divisors. An
  ideal is called maximal {\bf максимальным} if, moreover, the
  quotient is a field.
\end{opredelenie}

\begin{zadacha}
Prove that any prime ideal in an Artinian ring is maximal.
\end{zadacha}

\begin{zadacha}[*]
Describe all maximal ideals in the ring of polynomials $k[t]$.
\end{zadacha}

\begin{zadacha}
Consider the set of all nilpotent elements in the ring  $R$. Prove
that it is an ideal.
\end{zadacha}

\begin{opredelenie}
This ideal is called the  {\bf nilradical} of the ring $R$.
\end{opredelenie}

\begin{zadacha}[!]
Consider the quotient ring  $R/{\mathfrak n}$ of a ring by its
nilradical. Prove that $R/{\mathfrak n}$ has no nilpotent elements. 
\end{zadacha}

\begin{zadacha}
Consider an ideal in an Artinian ring that does not coincide with the
whole ring. Prove that it is contained in a maximal one.
\end{zadacha}

\begin{zadacha}[*] Consider an ideal in a ring (not necessary
  Artinian) that does not coincide with the whole ring. Prove that it
  is contained in a maximal one.
\end{zadacha}

\begin{ukazanie}
Use Zorn's lemma.
\end{ukazanie}

\begin{opredelenie}
An Artinian ring $R$ is called  {\bf semisimple},
if it does not have non-zero nilpotents.
\end{opredelenie}

\begin{opredelenie}
Consider a direct sum $\oplus R_i$ with the natural (coordinate-wise)
multiplication and addition. The resulting algebra is called
the {\bf direct sum of $R_i$} and is denoted $\oplus R_i$ too.
\end{opredelenie}

\begin{zadacha} 
Prove that the direct sum of semisimple Artinian rings in semisimple.
\end{zadacha}

\begin{zadacha}
Let $v$ be an element of a finite-dimensional algebra $R$ over $k$.
Consider a subspace $R$ generated by
$1, v, v^2, v^3, \dots$ (for all powers of $v$). 
Supposed this space has dimension $n$. Prove that $P(v)=0$ for some
polynomial $P= t^{n+1} + a_n t^n + \dots$
with coefficients in  $k$. Prove that this polynomial is unique.
\end{zadacha}

\begin{opredelenie}
This polynomial is called the  {\bf minimal polynomial} of the element $v$
and is denoted $\minpoly(v)$.
\end{opredelenie}

\begin{zadacha}
  Let $v\in R$ be an element of an Artinian ring over $k$, and $P(t)$
  be its minimal polynomial. Рассмотрим подалгебру $R_v$, порожденную
  $v$ и $k$. Докажите, что $R_v$ изоморфно кольцу $k[t]/P$ остатков по
  модулю $P$.
\end{zadacha}

\begin{opredelenie}
Let $v\in R$ be an element of an algebra $R$ such that $v^2=v$.
Then $v$ is called an {\bf idempotent}.
\end{opredelenie}

\begin{zadacha}
Let  $e\in R$ be an idempotent in a ring.
Prove that $1-e$ is an idempotent too. Prove that a product of
idempotents is and idempotent.
\end{zadacha}

\begin{zadacha}
Let $e\in R$ be an idempotent in a ring.
Consider the space $eR\subset R$
(the image of the multiplication by  $e$). Prove that $eR$ is a
subalgebra in  $R$, that $e$ is an identity in $eR$, and that $R=eR
\oplus (1-e)R$. 
\end{zadacha}

\begin{zadacha}[!]
Let $R = k(t)/P$ where $P$ is a polynomial that decomposes into a 
product of mutually co-prime polynomials $P = P_1 P_2 \dots
P_n$. Prove that  $R$ has $m$ idempotents $e_1, \dots, e_n \subset R$,
and that $e_i R \cong k[t]/P_i$.
\end{zadacha}

\begin{ukazanie}
Find polynomials  $Q(t)$, $Q'(t)$ such that
$Q P_1 + Q' P_1 P_3 \dots P_n=1$. Let
$e = Q'  P_1 P_3 \dots P_n$. Prove that
$e^2 = e (\mod P)$, и $eP_1(t)=0 (\mod P)$.
Deduce that $k[z]/P_1(z) \cong e R$, and the isomorphism
is given by  $z \mapsto et$.
\end{ukazanie}

\begin{zadacha}
Let  $R$  be a semisimple Artinian ring without non-identity
idempotents. Prove that it is a field.
\end{zadacha}

\begin{ukazanie}
Let  $R$ be a field. Consider the subalgebra  $k(x) \subset R$
generated by a non-invertible element $x \in R$, and apply the
previous problem.
\end{ukazanie}

\begin{opredelenie}
Two idempotents $e_1,e_2 \in R$ in a commutative algebra  $R$ are
called {\bf orthogonal} if $e_1e_2=0$.
\end{opredelenie}

\begin{zadacha} Let $e_1,e_2,e_3 \in R$ be idempotents in an Artinian ring
$R$ over a field $k$ and let $e_1=e_2+e_3$, let $e_2$ and $e_3$ be
orthogonal. Prove that $e_2,e_3 \in e_1R$ and $e_1R=e_2R \oplus
e_3R$. 
\end{zadacha}

\begin{zadacha} 
Let $\cchar k \neq 2$. Suppose that $e_1, e_2, e_3$ be idepmotents in
an Artinian ring $R$ over a ring $k$ and $e_1 = e_2 + e_3$. 
Prove that $e_2$ and $e_3$ are orthogonal.
\end{zadacha}

\begin{opredelenie} Let $R$ be an Artinian ring over a field $k$.
An idempotent $e$ in $R$ is called
{\bf indecomposable} if there are no such non-zero orthogonal
idempotents  $e_2, e_3$ such that $e_1 = e_2 + e_3$.
\end{opredelenie}

\begin{zadacha}[!]
Let  $R$ be a semisimple Artinian ring and $e$ be an indecomposable
idempotent. Prove that $eR$ is a ring.
\end{zadacha}

\begin{zadacha}[!]
Let  $R$ be a semisimple Artinian ring over a field $k$.
Prove that $1$ decomposes into a sum of indecomposable orthogonal
idempotents: $1 = \sum e_i$. Prove that this decomposition is unique. 
\end{zadacha}

\begin{ukazanie}
For existence take some idempotent $e \in R$ and decompose
 $R=eR \oplus (1-e)R$ then use induction. For uniqueness, consider the
 product of two possible decompositions of $1$.
\end{ukazanie}

\begin{zadacha}[!]
Let $R$ be a semisimple Artinian ring over a ring $k$. Prove that  $R$
is isomorphic to  a direct sum of fields.
\end{zadacha}

\begin{ukazanie}
Use the previous problem.
\end{ukazanie}

\begin{zadacha}[!]
Let  $R_1 \overset{\psi}{\arrow} R_2$ be a surjective homomorphism of
Artinian rings,  moreover, let $R_1$ be semisimple and thus decomposed
into a direct sum of fields over some set of indices $I$, 
$R_1= \oplus_{i\in I} K_i$. Prove that $R_2 = \oplus_{i\in I'} K_i$,
where $I'$ is some subset of $I$ and $\psi$ is the natural projection
(that is, $\psi$ acts identically on
$K_i$, $i \in I'$ and is zero on $K_i$, $i \notin I'$).
\end{zadacha}

\begin{ukazanie}
Decompose  $1 \in R_1$ into the sum of indecomposable idempotents $e_i$, $i \in I$,
prove that $f:e_iR \to f(e_i)R_2$ is surjective for all $i \in I$,
and apply Problem~\ref{field.noideals}.
\end{ukazanie}

\begin{zadacha}[*]
  Let $R=k[t]/P$ and the polynomial $P$ has multiple roots over the
  algebraic closure $\bar k$. Can $R$ be semisimple? Analyse the cases
  $\cchar k =0$, $\cchar k \neq 0$.
\end{zadacha}

\begin{zadacha}[*]
Let  $R$ be a semisimple Artinian ring over a field $k$, and  $1 =
e_1+ \dots + e_n$ be the decomposition of 1 into the sum of
indecomposable orthogonal idempotents. Prove that $R$ has exactly  $n$
prime ideals. Describe these ideals in terms of $e_i$.
\end{zadacha}

\begin{zadacha}[*]
Let  $R$ be an Artinian ring over a field $k$ (of any characteristic). 
Prove that the intersection of all simple ideals  $R$ is the
nilradical of $R$.
\end{zadacha}

\begin{opredelenie}
Let $R$ be an algebra over a field $k$,
and  $g$ be a bilinear form on  $R$. The form $g$ is called
{\bf invariant}, if  $g(x, yz) = g(xy, z)$ for any  $x$, $y$, $z$.
\end{opredelenie}

\begin{zadacha} 
Let  $R$ be an Artinian ring endowed with a bilinear invariant form,
and ${\mathfrak m}$ be an ideal in $R$.
Prove that ${\mathfrak m}^\bot$ is an ideal too.
\end{zadacha}

\begin{zadacha}[*]
Find an Artinian ring that does not admit a non-degenerate invariant
bilinear form.
\end{zadacha}

\begin{zadacha}[!] \label{_Tr_semisimple_Zadacha_}
Let  $R$ be an Artinian ring over a field $k$.
Consider a the bilinear form $a, b \arrow \tr(ab)$,
where $\tr(ab)$ is the trace of the endomorphism
$L_{ab}\in \End R$, $x \stackrel {L_{ab}}\mapsto abx$.
Prove that if this form is non-degenerate then  $R$ is semisimple. 
Prove that if  $R$ is semisimple and $\cchar k =0$ then the form is
non-degenerate.
\end{zadacha}

\begin{ukazanie}
One direction can be proved using the Problem~\ref{tr=0}. For the
other direction consider first the situation when $R$ is a field. 
\end{ukazanie}

\begin{zadacha} 
Let $V$, $V'$ be vector spaces over $k$ endowed with bilinear forms
$g$, $g'$. Define on $V\otimes V'$ the bilinear form $g \otimes g'$
that would satisfy  
\[ g \otimes g'(v\otimes v',w\otimes w')= g(v,w)g'(v', w')
\]
Prove that the bilinear form on $V \otimes V'$ is well-defined and
unique.
\end{zadacha}

\begin{zadacha} 

Let  $R$, $R'$ be commutative algebras over  $k$. Consider a tensor
product  $R \otimes R'$. Endow  $R \otimes R'$ with a multiplicative
structure such that  $v\otimes v' \cdot w\otimes w = vw\otimes v'w'$.
Prove that the ring structure on $R \otimes R'$  is well-defined and
unique.
\end{zadacha}

\begin{zadacha}
Describe the algebra $\C \otimes_\R \C$.
\end{zadacha}

\begin{zadacha}
Describe the algebra $\Q[\1]\otimes_\Q \Q[\1]$.
\end{zadacha}
and apply the problem
\begin{zadacha}[!]
Let $P(t)$ and $Q(t)$ be polynomials over a field $k$. Denote
$K_1=k[t]/P(t)$ and $K_2=k[t]/Q(t)$. Prove that 
$K_1 \otimes K_2 \cong K_1[t]/Q(t) \cong K_2[t]/P(t)$.
\end{zadacha}

\begin{zadacha}[*]
Let  $R$, $R'$ be Artinian rings over $k$, $\cchar k =0$. Denote the
natural bilinear forms $a, b \arrow \tr(ab)$ on these rings by  $g$,
$g'$. Consider the tensor product  $R \otimes R'$ with the natural
structure of Artinian algebra. Consider the form  $g\otimes g'$ on $R
\otimes R'$. Prove that $g\otimes g'$ is equal to the form $a, b
\arrow \tr(ab)$. 
\end{zadacha}

\begin{zadacha}[*]
Prove that the tensor product of semisimple Artinian rings over a
field  $k$ of characteristic $0$ is semisimple.
\end{zadacha}

\begin{ukazanie}
Use the Problem~\ref{_Tr_semisimple_Zadacha_}.
\end{ukazanie}

\begin{zadacha}[*]
Find two fields $K_1$, $K_2$, algebraic over but not equal to $\Q$,
such that $K_1\otimes_\Q K_2$ is also a field.
\end{zadacha}

\begin{zadacha}[*]
Let  $P(t)\in \Q[t]$ be a polynomial that does not have rational roots
but has exactly $r$ real and  $2s$ complex roots (that are
non-real). Prove that 
\[ (\Q[t]/P)\otimes_\Q \R = \bigoplus_s \C \oplus \bigoplus_r \R.
\]
\end{zadacha}

\begin{zadacha}[*]
Let  $P(t)$ be an irreducible polynomial over $\Q$ that does not have
real roots and  $v\in \Q[t]/P$ be an element that does not belong to
$Q \subset \Q[t]/P$. Prove that the minimal polynomial of  $v$ does
not have real roots.
\end{zadacha}

\end{document}