Sentence examples for module of length from inspiring English sources

Conversely, assume that X is a strongly regular module of length | X | = d.

The Γ ( C ) -module Hom ( C, Y ) is the indecomposable projective module of length 3.

When dealing with a local uniserial ring, the indecomposable module of length n will be denoted just be n.

There is the following converse: Let Q = ( D Λ ) b. Then any module of length at most b is present in Λ [ → Q 〉. Let M be a module of length at most b, thus the socle soc M of M is a semisimple module of length at most b and therefore a submodule of Q = ( D Λ ) b. It follows that M itself can be embedded into Q.

Every summable module M is a Σ-module and every totally projective module of length ω+1 is a direct sum of countably generated modules.

A module M is summable and a strong ω-elongation of a totally projective module by a (ω+1 -projective module if and only if M is a totally projective module of length ≤ω+1 -projective

Let R ( i ) be the set of isomorphism classes of strongly regular modules of length i.

For ( 3, 3 ), the module X must be a strongly regular module, thus, there are three different kinds: direct sums of three pairwise non-isomorphic indecomposable modules of length 2, direct sums of an indecomposable module M of length 4 and an indecomposable module M ′ of length 2 such that Hom ( M, M ′ ) = 0, and finally indecomposable modules of length 6.

Now assume that (n > 1) and the result is true for all modules of length less than (n).

The regular Kronecker modules of length 2 different from R ∞ will be denoted by R λ with λ ∈ k.

In [5], it was shown that any ( ω + 1 ) -projective σ -module is a direct sum of countable modules of length atmost ( ω + 1 ).