Kernels of Covered Groups

The lattice of ideals of MR(R2)Ra commutative pir

“…Then 1 = e x + e 2 -\ h e t where {e,} is a set of orthogonal idempotents, e t the identity of 5 ( . Note further that S 2 = S 2 ©..-©S 2 and so, from the previous theorem, to determine the ideals of M S (S ) it suffices to determine the ideals of the individual components.…”

“…Moreover, when R is a principal ideal domain, there is a lattice isomorphism between the ideals of R and the lattice of two-sided [2] The lattice of ideals 369 invariant subgroups of M R (R 2 ). In this work we turn to the case in which R is a commutative principal ideal ring and investigate the lattice of ideals of M R (R 2 ).…”

“…Thus we see that the collection of submodules {H(P, a { , ... , <*",_,) \ P e F, a , , ... , a m -1 € F} is a cover for i? 2 (see [2]) and we call the submodules H(P, a , , ... , a m _ l ) covering submodules. Therefore, to define a function / in N it suffices to define / on the generators of the covering submodules, use the homogeneous property f{xr) = f(x)r to extend / to all of R 2 and then verify that / is well-defined.…”

“…Our work also has geometric connections. Specifically, let R be a principal ideal ring and let W be a cover (see [2]) of R 2 by cyclic submodules. Then for each / e M R (R 2 ) and each W a e W, there exists ^ € & such that…”

“…Unless otherwise stated, in this section R will denote a special PIR with unique maximal ideal J = {8} and index of nilpotency m. Let / be an ideal of N = M R (R 2 ) with / C (A 2 k :R 2 ) as given in Proposition 2.3. From the fact that Jf 2 {A k ) C / our plan is to show that an arbitrary function in (A 2 k :R 2 ) can be constructed from functions in / . This will then give the desired equality.…”