Fuzzy chain subrings and their fuzzy direct sum
In this paper the author characterizes chain rings in terms of certain fuzzy ideals by using minimal generating sets. He employs the concept of fuzzy direct sum of fuzzy subrings to characterize Artinian principal ideal rings and to find necessary and sufficient conditions under which an Artinian ring is a direct sum of fields. Let \(R\) be a commutative ring with identity. \(R\) is called a chain ring if its ideals form a chain. A fuzzy ideal \(\mu\) of \(R\) is called a fuzzy chain if for all \(x, y\) in \(R\), (1) \(\mu(x)= \mu(y)\) if and only if the ideal \((x) =\) the ideal \((y)\), and (2) \(\mu(x)< \mu(y)\) if and only if \((x)\supset (y)\). The author shows that \(R\) is a chain ring if and only if \(R\) has fuzzy ideal which is a fuzzy chain. The author also shows that the following conditions are equivalent on a fuzzy ideal of an Artinian ring \(R\): (1) \(\mu\) is a fuzzy chain, (2) the level ideal \(\mu_{\mu(x)}\) equals the ideal generated by \(x\) for any \(x\in R\), (3) \(\mu\) is fuzzy local, fuzzy principal, and of maximal chain.
scientific article; zbMATH DE number 1375041
scientific article