MATHEMATICS
Conjugate idempotent formal matrices of order $2$ over residue class rings
A. E. Zykov,
A. M. Koroleva,
Ts. D. Norbosambuev Tomsk State University, Tomsk, Russian Federation
Abstract:
Let
$p$ be a prime,
$p > 1$,
$m$ and
$n$ be integers,
$m > n > 0$. In recent works [1–5]
the following formal matrix rings were considered:
$$
K =
\begin{pmatrix}
\mathbf{Z} /p^m\mathbf{Z} & \mathbf{Z}/ p^n\mathbf{Z}\\
\mathbf{Z} /p^n\mathbf{Z} & \mathbf{Z}/ p^n\mathbf{Z}
\end{pmatrix}
=\left\{
\begin{pmatrix}
a+p^m\mathbf{Z} & b+ p^n\mathbf{Z}\\
c+p^n\mathbf{Z} & d+ p^n\mathbf{Z}
\end{pmatrix}
\mid a, b, c, d\in\mathbf{Z}
\right\}
$$
with multiplication defined so that for every
$A, A'\in K$ we have
\begin{multline*}
A\cdot A'=
\begin{pmatrix}
a+p^m\mathbf{Z} & b+ p^n\mathbf{Z}\\
c+p^n\mathbf{Z} & d+ p^n\mathbf{Z}
\end{pmatrix}
\begin{pmatrix}
a'+p^m\mathbf{Z} & b'+ p^n\mathbf{Z}\\
c'+p^n\mathbf{Z} & d'+ p^n\mathbf{Z}
\end{pmatrix}
=\\
=
\begin{pmatrix}
aa'+p^{m-n}bc'+p^m\mathbf{Z} & ab'+bd'+ p^n\mathbf{Z}\\
ca'+dc'+p^n\mathbf{Z} & p^{m-n}cb' +dd'+ p^n\mathbf{Z}
\end{pmatrix}.
\end{multline*}
It is known [2–5] that the matrix $A=\begin{pmatrix}
a+p^m\mathbf{Z} & b+ p^n\mathbf{Z}\\
c+p^n\mathbf{Z} & d+ p^n\mathbf{Z}
\end{pmatrix}\in K
$ is nilpotent (invertible) if and only if
$p$ divides (does not divide)
$a$ and
$d$.
In [1] it was shown that
$A$ is a nontrivial idempotent in
$K$ if and only if
$A$ has the form
$$
\begin{pmatrix}
1-\sigma_{\nu+1}+p^m\mathbf{Z} & b+ p^n\mathbf{Z}\\
c+p^n\mathbf{Z} & \sigma+ p^n\mathbf{Z}
\end{pmatrix}\quad
\text{ or }\quad
\begin{pmatrix}
\sigma_{\nu+1}+p^m\mathbf{Z} & b+ p^n\mathbf{Z}\\
c+p^n\mathbf{Z} & 1-\sigma+ p^n\mathbf{Z}
\end{pmatrix}
$$
where
$b$,
$c \in \mathbf{Z}$, $\sigma=\sum\limits_{k=1}^{\nu+1} C_k(p^{m-n}bc)^k$,
$\nu = \left[\frac{n-1}{m-n}\right]$ and
$C_i$ are Catalan numbers. For
every
$i > 0$ we define the
$i$th Catalan number by $C_i=\frac1i\binom{2i-2}{i-1}=\frac{(2i-2)!}{i!(i-1)!}$, so
$C_1 = 1$,
$C_2 = 1$,
$C_3 = 2$,
$C_4 = 5$,
$C_5 = 14$, etc.
Let us call a non-trivial idempotent matrix with an invertible element in the upper left
corner an
idempotent matrix of type $1$. An
idempotent matrix of type $2$ is a non-trivial
idempotent matrix with an invertible element in the lower right corner.
Definition 2.1. Idempotent elements
$e_1$ and
$e_2$ of ring
$R$ are
conjugate if there is an invertible
element
$u\in R$ such that
$e_2 = ue_1u^{–1}$.
We have obtained the following results.
Theorem 2.3. In the formal matrix ring
$K$ every idempotent matrix of type
$1$ is conjugate
to the matrix $E_{11}=\begin{pmatrix}
1+p^m\mathbf{Z} & 0+ p^n\mathbf{Z}\\
0+p^n\mathbf{Z} & 0+ p^n\mathbf{Z}
\end{pmatrix}
$.
Likewise, every idempotent matrix of type
$2$ is
conjugate to the matrix
$E_{22}=\begin{pmatrix}
0+p^m\mathbf{Z} & 0+ p^n\mathbf{Z}\\
0+p^n\mathbf{Z} & 1+ p^n\mathbf{Z}
\end{pmatrix}
$.
Corollary 2.4. In the formal matrix ring
$K$ two idempotent matrices of different types are
never conjugate.
Corollary 2.6. In the formal matrix ring
$K$ any two idempotent matrices of the same type
are conjugate.
Keywords:
formal matrix ring, idempotent formal matrix, conjugate idempotents.
UDC:
512.552
MSC: 16S50 Received: 16.03.2025
Accepted: June 9, 2025
DOI:
10.17223/19988621/95/2
Fulltext:
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