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Projections of finite rings
S. S. Korobkov Urals State Pedagogical University, Ekaterinburg
Abstract:
Let
$R$ and
$R^{\varphi}$ be associative rings with isomorphic subring lattices, and
$\varphi$ be a lattice isomorphism (or else a projection) of the ring
$R$ onto the ring
$R^{\varphi}$. We call
$R^{\varphi}$ the projective image of a ring
$R$ and call
$R$ itself the projective preimage of a ring
$R^{\varphi}$. The main result of the first part of the paper is Theorem 5, which proves that the projective image
$R^{\varphi}$ of a one-generated finite
$p$-ring
$R$ is also one-generated if
$R^{\varphi}$ at the same time is itself a
$p$-ring. In the second part, we continue studying projections of matrix rings. The main result of this part is Theorems 6 and 7, which prove that if
$R=M_n(K)$ is the ring of all square matrices of order
$n$ over a finite ring
$K$ with identity, and
$\varphi$ is a projection of the ring
$R$ onto the ring
$R^{\varphi}$, then
$R^{\varphi}=M_n(K')$, where
$K'$ is a ring with identity, lattice-isomorphic to the ring
$K$.
Keywords:
one-generated finite rings, matrix rings, lattice isomorphisms of associative rings.
UDC:
512.552 Received: 19.01.2023
Revised: 19.07.2024
DOI:
10.33048/alglog.2023.62.405
Fulltext:
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