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MATHEMATICS
Idempotent and nil-clean formal matrices of order 2 over residue class rings
A. M. Korolevaa,
Ts. D. Norbosambueva,
M. V. Podkorytovb a Tomsk State University, Tomsk, Russian Federation
b Novosibirsk State University, Novosibirsk, Russian Federation
Abstract:
Let us consider the set of formal matrices
$$
K =
\begin{pmatrix}
\mathbf{Z} /p^m\mathbf{Z} & \mathbf{Z}/ p^n\mathbf{Z}\\
\mathbf{Z} /p^n\mathbf{Z} & \mathbf{Z}/ p^n\mathbf{Z}
\end{pmatrix}
=\left\{
\begin{pmatrix}
a+p^m\mathbf{Z} & b+ p^n\mathbf{Z}\\
c+p^n\mathbf{Z} & d+ p^n\mathbf{Z}
\end{pmatrix}
\mid a, b, c, d\in\mathbf{Z}
\right\}
$$
over residue class rings
$\mathbf{Z}/p^m\mathbf{Z}$ and
$\mathbf{Z}/p^n\mathbf{Z}$,
where
$p$ is a prime,
$m$ and
$n$ are natural numbers,
$m > n > 0$. For any matrices
$A, A'\in K$, we define multiplication as follows:
\begin{multline*}
\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*}
With entrywise addition and multiplication introduced as above, the set
$K$ forms a ring.
It is known that a formal pmatrix from
$K$ is invertible if and only if its elements on the main diagonal are not multiples of
$p$.
It is also known that a formal pmatrix from
$K$ is nilpotent if and only if its elements on the main diagonal are multiples of
$p$.
We call the sequence of natural numbers defined as follows Catalan numbers:
$C_1 = 1$,
$C_{k+1} =\sum\limits_{i=1}^k C_iC_{k-i+1}$, i.e.
$C_1 = 1$,
$C_2 = 1$,
$C_3 = 2$,
$C_4 = 5$,
$C_5 = 14$,
$C_6 = 42$ and so on.
The following theorem is the main result of our paper.
Theorem 1.2. Matrix
$A$ is a nontrivial idempotent of
$K$ if and only if
$A$ has the form
$$
A=
\begin{pmatrix}
1-\sigma_{\nu+1}+p^m\mathbf{Z} & b+ p^n\mathbf{Z}\\
c+p^n\mathbf{Z} & \sigma_\nu+ p^n\mathbf{Z}
\end{pmatrix}\quad
\text{ or }\quad
A=
\begin{pmatrix}
\sigma_{\nu+1}+p^m\mathbf{Z} & b+ p^n\mathbf{Z}\\
c+p^n\mathbf{Z} & 1-\sigma_\nu+ p^n\mathbf{Z}
\end{pmatrix}
$$
where
$b$,
$c \in \mathbf{Z}$,
$\nu = \left[\frac{n-1}{m-n}\right]$, $\sigma_\nu=\sum\limits_{k=1}^\nu C_k(p^{m-n}bc)^k$, where
$C_i$ are Catalan numbers.
Note that
$\nu$ here can be equal to
$0$ (then
$\sigma_\nu = 0$).
Corollary 2.2. The ring
$K$ is not a
$k$-nil-clean ring for any natural number
$k$ such that
$k < p-1$.
Keywords:
formal matrix ring, idempotent formal matrix, nil-clean ring, Morita context ring.
UDC:
512.552
MSC: 08A35,
15B99,
16S50 Received: 15.10.2024
Accepted: February 7, 2025
DOI:
10.17223/19988621/93/3
Fulltext:
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