Forcing Linearity Numbers for Finitely Generated Modules

Forcing linearity numbers for coatomic modules

“…The following example shows that Theorem 2.3 is not true in general, if I is not the annihilator of some 0 = w ∈ V. Proof. It is well-known that there exist coatomic modules V over a commutative ring R such that f ln R (V ) ∈ {0, 1, 2, ∞}, see for example [5]. If V is a cyclic module, then M R (V ) = End R (V ), hence f ln R (V ) = 0.…”

“…Forcing linearity numbers have been found for several classes of rings and modules, see for example [3], [4], [5] and their references. In section 2 we determine the forcing linearity number of coatomic modules over an arbitrary commutative ring R with identity.…”

“…Then n is the forcing linearity number of a coatomic module over a commutative ring if and only if n ∈ {0, 1, 2, ∞} ∪ {q + 2| q is a prime power}.Proof. It is well-known that there exist coatomic modules V over a commutative ring R such that f ln R (V ) ∈ {0, 1, 2, ∞}, see for example[5]. If V is a cyclic module, thenM R (V ) = End R (V), hence f ln R (V ) = 0.…”