A few years ago, I defined a squarefree module over a polynomial ring S = k[x1, . . . , xn] generalizing the Stanley-Reisner ring k[∆] = S/I∆ of a simplicial complex ∆ ⊂ 2. This notion is very useful in the StanleyReisner ring theory. In this paper, from a squarefree S-module M , we construct the k-sheaf M on an (n − 1) simplex B which is the geometric realization of 2. For example, k[∆] is (th...

A number of results are proved concerning the Quillen ^-theory K+(S*G) of the skew group ring S*G, where S is a Noetherian ring and G is a finite group of automorphisms of 5. Applications are given to the computation of AT-groups of group algebras and of equivariant /^-theory for affine varieties.

the zero-divisor graph of a commutative ring r with respect to nilpotent elements is a simple undirected graph $gamma_n^*(r)$ with vertex set z_n(r)*, and two vertices x and y are adjacent if and only if xy is nilpotent and xy is nonzero, where z_n(r)={x in r: xy is nilpotent, for some y in r^*}. in this paper, we investigate the basic properties of $gamma_n^*(r)$. we discuss when it will be eu...

Let $R$ be a ring and $M$ a right $R$-module with $S=End_R(M)$. A module $M$ is called semi-projective if for any epimorphism $f:Mrightarrow N$, where $N$ is a submodule of $M$, and for any homomorphism $g: Mrightarrow N$, there exists $h:Mrightarrow M$ such that $fh=g$. In this paper, we study SGQ-projective and $pi$-semi-projective modules as two generalizations of semi-projective modules. A ...

The title compound, C(21)H(29)ClO(3) [systematic name (8R,9S,10R,13S,14S,17S)-4-chloro-3-oxoandrost-4-en-17β-yl acetate], is a 4-chloro derivative of testosterone, used as an anabolic androgenic agent or applied topically in ophthalmological and dermatological treatments. The absolute configurations at positions 8, 9, 10, 13, 14 and 17 were established by refinement of the Flack parameter as R,...

A few years ago, I defined a squarefree module over a polynomial ring S = k[x1, . . . , xn] generalizing the Stanley-Reisner ring k[∆] = S/I∆ of a simplicial complex ∆ ⊂ 2. This notion is very useful in the StanleyReisner ring theory. In this paper, from a squarefree S-module M , we construct the k-sheaf M on an (n − 1) simplex B which is the geometric realization of 2. For example, k[∆] is (th...

 prsnt img-s rnring tht llows r viwpoint wlkthroughs o urn sns using just  w photogrphs s input. Commril pplitions suh s Googl trtviw, Bing ps t. us ruimntry orms o img-s rnring or urn visuliztion; mor sophistit pprohs us th ull D mol o th sn s input. As th qulity o Dmol grs, rnring rtits r...

A skew generalized power series ring R[[S, ω]] consists of all functions from a strictly ordered monoid S to a ring R whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action ω of the monoid S on the ring R. Special cases of the skew generalized power series ring construction are...

In the title compound, C(22)H(19)N(5)OS·H(2)O, the naphthalene ring system and the benzene ring [dihedral angle = 85.19 (8)°] make dihedral angles of 87.02 (9) and 14.41 (10)°, respectively, with the pyrazole ring. The mean plane through the 2-methyl-enehydrazinecarbothio-amide group [C-N-N-C(=S)-N; maximum deviation = 0.022 (1) Å] is slightly twisted from the pyrazole ring [dihedral angle = 5....

Inducing patterns that selectively stimulate the S cones can induce large shifts in color appearance. For example, a "peach" test-ring presented within contiguous purple and non-contiguous lime inducing rings appears pink while the physically identical peach test-ring appears orange when presented within contiguous lime and non-contiguous purple inducing rings (Fig. 1c). These shifts have been ...