Projections of finite one-generated rings with identity

S. S. Korobkov

Algebra i logika, Tome 55 (2016) no. 2, pp. 192-218

Résumé

Associative rings $R$ and $R'$ are said to be lattice-isomorphic if their subring lattices $L(R)$ and $L(R')$ are isomorphic. An isomorphism of the lattice $L(R)$ onto the lattice $L(R')$ is called a projection (or else a lattice isomorphism) of the ring $R$ onto the ring $R'$. A ring $R'$ is called the projective image of a ring $R$. Lattice isomorphisms of finite one-generated rings with identity are studied. We elucidate the general structure of finite one-generated rings with identity and also give necessary and sufficient conditions for a finite ring decomposable into a direct sum of Galois rings to be generated by one element. Conditions are found under which the projective image of a ring decomposable into a direct sum of finite fields is a one-generated ring. We look at lattice isomorphisms of one-generated rings decomposable into direct sums of Galois rings of different types. Three main types of Galois rings are distinguished: finite fields, rings generated by idempotents, and rings of the form $GR(p^n,m)$, where $m>1$ and $n>1$. We specify sufficient conditions for the projective image of a onegenerated ring decomposable into a sum of Galois rings and a nil ideal to be generated by one element.

Détail

Keywords: finite rings, one-generated rings, lattice isomorphisms of associative rings.