Ultraproducts of pm-rings and mp-rings

Ultraproducts of Group Rings

Group Rings Let G = g1, g2, . . . , gn be a finite group, and let k be a field. We define the group ring k[G] to be the set of sums of the form a1g1 + a2g2 + · · ·+ angn with each ai ∈ k and gi ∈ G. Addition is defined componentwise, i.e. (a1g1 + a2g2 + · · ·+ angn) + (b1g1 + b2g2 + · · ·+ bngn) = ((a1 + b1)g1 + (a2 + b2)g2 + · · ·+ (an + bn)gn). We define multiplication in the following way: (...

Ultraproducts of Noetherian Local Rings: Properties and Applications

The present proposal is in part a continuation of the NSF Research Proposal Definability, Constructibility and Transfer (DMS-0100778/DMS-0302248 in the amount of $69,276; project details are given in §4.3 and the introduction to §5). Whereas the latter proposal was submitted via the division Foundations, in view of the underlying tools stemming from model-theory, the present proposal is cast en...

Differential Polynomial Rings of Triangular Matrix Rings

On n-coherent rings, n-hereditary rings and n-regular rings

We observe some new characterizations of $n$-presented modules. Using the concepts of $(n,0)$-injectivity and $(n,0)$-flatness of modules, we also present some characterizations of right $n$-coherent rings, right $n$-hereditary rings, and right $n$-regular rings.

$n$-cocoherent rings‎, ‎$n$-cosemihereditary rings and $n$-V-rings

 Let $R$ be a ring‎, ‎and let $n‎, ‎d$ be non-negative integers‎. ‎A right $R$-module $M$ is called $(n‎, ‎d)$-projective if $Ext^{d+1}_R(M‎, ‎A)=0$ for every $n$-copresented right $R$-module $A$‎. ‎$R$ is called right $n$-cocoherent if every $n$-copresented right $R$-module is $(n+1)$-coprese-nted‎, ‎it is called a right co-$(n,d)$-ring if every right $R$-module is $(n‎, ‎d)$-projective‎. ‎$R$...