2,5,10,13-Tetraazatricyclo [12,2,2,26,9] icosa-1(16),6,8,14,17,19-hexene-3,4,11,12-tetraone tetraoxime (1a) and 2,5,11,14-tetraazatricyclo [13,3,1,16,10] icosa-1(19),6,8,10(20),15,17-hexaene-3,4,12,13-tetraonetetraoxime (1b) were obtained from condensation of 1,4- and 1,3-phenylendiamine with dichloroglyoxime, respectively. The reaction of 1,2-phenylendiamine with dich...

Let R be a commutative ring. An R-module M is called co-multiplication provided that foreach submodule N of M there exists an ideal I of R such that N = (0 : I). In this paper weshow that co-multiplication modules are a generalization of strongly duo modules. Uniserialmodules of finite length and hence valuation Artinian rings are some distinguished classes ofco-multiplication rings. In additio...

In the title compound, C(21)H(28)O(4)·H(2)O, the cyclo-hexa-dienone ring is planar (r.m.s. deviation 0.0186 Å), whereas the two cyclo-hexane rings and the cyclo-hexa-none ring adopt chair conformations. The crystal structure is stabilized by O-H⋯O and C-H⋯O hydrogen bonds.

In the title mol-ecule, C(30)H(32)N(2)O(2), the two tolyl rings form dihedral angles of 65.8 (1) and 6.6 (1)° with the isoindole-1,3-dione mean plane. The cyclo-hexane ring adopts a chair conformation.

In title compound, C(27)H(36)O, all cyclo-hexane rings within the adamantyl groups adopt chair conformations. There are no obvious inter-molecular hydrogen bonds in the structure, so that van der Waals attractions stabilize the crystal.

In the title compound, C(19)H(33)NO, all three cyclo-hexane rings adopt chair conformations. The crystal packing features weak C-H⋯O inter-actions, forming a supra-molecular chain along the c axis.

Let $A$ be a commutative ring with nonzero identity, and $1leq n

Let $R$ be a commutative ring with identity. A proper submodule $N$ of an $R$-module $M$ is an n-submodule if $rmin N~(rin R, min M)$ with $rnotinsqrt{Ann_R(M)}$, then $min N$. A number of results concerning n-submodules are given. For example, we give other characterizations of n-submodules. Also various properties of n-submodules are considered.

in this note we introduce the notion of weak mccoy rings as a generalization of mccoy rings, and investigate their properties. also we show that, if is a semi-commutative ring, then is weak mccoy if and only if is weak mccoy.