for each  and i ≥ 0. The polynomial ring of integer-valued in rational  is defined by Int (    an important example binomial and is non-Noetherian ring. In this paper the algebraic structure rings has been studied their properties ideals. notion ideal generated a given set defined.  Which allows us to define new class Noetherian using ideals, which we named it binomiall...

In this paper, we consider the openness of P-locus a finitely generated module over commutative noetherian ring in case where P is each properties FID, Gor, Can, CM, MCM, (Sn), and (Tn). One main results asserts that FID-loci an acceptable are open. We give version Nagata criterion, prove it holds for all aforementioned properties.

We describe new classes of noetherian local rings R whose finitely generated modules M have the property that ToriR(M,M)=0 for i≫0 implies has finite projective dimension, or ExtRi(M,M)=0 dimension injective dimension.