Associative rings are considered. By a lattice isomorphism, or projection, of a ring R onto a ring R φ we mean an isomorphism φ of the subring lattice L(R) of R onto the subring lattice L(R φ ) of R φ . In this case R φ is called the projective image of a ring R and R is called the projective preimage of a ring R φ . Let R be a finite ring with identity and Rad R the Jacobson radical of R. A ring R is said to be local if the factor ring R/ Rad R is a field. We study lattice isomorphisms of finite local rings. It is proved that the projective image of a finite local ring which is distinct from GF( p q n $$ {\mathrm{p}}^{{\mathrm{q}}^{\mathrm{n}}} $$ ) and has a nonprime residue field is a finite local ring. For the case where both R and R φ are local rings, we examine interrelationships between the properties of the rings.

中文翻译：

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有限局部环的格同构

考虑关联环。通过环 R 到环 R φ 的晶格同构或投影，我们指的是 R 的子环晶格 L(R) 到 R φ 的子环晶格 L(R φ ) 的同构 φ。在这种情况下，R φ 称为环 R 的投影图像，R 称为环 R φ 的投影原像。设 R 是一个具有身份的有限环，Rad R 是 R 的 Jacobson 根。如果因子环 R/Rad R 是一个域，则称环 R 是局部的。我们研究有限局部环的晶格同构。证明了不同于 GF( pqn $$ {\mathrm{p}}^{{\mathrm{q}}^{\mathrm{n}}} $$ ) 和有一个非素残差域是一个有限局部环。对于 R 和 R φ 都是局部环的情况，我们检查环的属性之间的相互关系。