We give an elementary description of the maps in the linear strand of the minimal free resolution of a square-free monomial ideal, that is, the Stanley-Reisner ideal associated to a simplicial complex. The description is in terms of the homology of the canonical Alexander dual complex. As applications we are able to prove for monomial ideals and j = 1 a conjecture of J. Herzog giving lower boun...

in this paper, we introduce the new notion of n-f-clean rings as a generalization of f-clean rings. next, we investigate some properties of such rings. we prove that mn(r) is n-f-clean for any n-f-clean ring r. we also, get a condition under which the denitions of n-cleanness and n-f-cleanness are equivalent.

a topoframe, denoted by $l_{ tau}$,  is a pair $(l, tau)$ consisting of a frame $l$ and a subframe $ tau $ all of whose elements are complementary elements in$l$. in this paper, we define and study the notions of a$tau $-real-continuous function on a frame $l$ and the set of realcontinuous functions $mathcal{r}l_tau $ as an $f$-ring.we show that $mathcal{r}l_{ tau}$is actually a generalization ...

for a fixed positive integer , we say a ring with identity is n-generalized right principally quasi-baer, if for any principal right ideal of , the right annihilator of is generated by an idempotent. this class of rings includes the right principally quasi-baer rings and hence all prime rings. a certain n-generalized principally quasi-baer subring of the matrix ring are studied, and connections...

Photoelectrocatalysis can directly store solar energy in chemical fuels like hydrogen, but its performance relies on external factors (such as light intensity and weather). In article number 2008182, Erwin Reisner co-workers report a hybrid perovskite-BiVO4 device that perform light-driven water splitting during daytime switch to electrolysis at night, or when renewable electricity is more abun...

The notion of strict closedness rings was given by J. Lipman [Amer. Math. 93 (1971), pp. 649–685] in connection with a conjecture O. Zariski. present purpose is to give practical method construction strictly closed rings. It also shown that the Stanley-Reisner simplicial complexes (resp. $F$-pure satisfying condition $(\operatorname {S}_2)$ Serre) are weakly Arf)