The title compound, C(26)H(18)Cl(2)O, is a heterocyclic structure consisting of a benzo[f]chromene ring and two aromatic rings. The non-H atoms of the benzo[f]chromene ring are almost coplanar (rms deviation = 0.107 Å), and the methyl C atom lies 1.340 (4) Å from the mean plane of the benzo[f]chromene ring. The chromene ring forms dihedral angles of 88.45 (2)° with the benzene ring linked to th...

Let R be an associative ring with identity, C(R) be the category of com-plexes of R-modules and Flat(C(R)) be the class of all at complexes of R-modules. We show that the at cotorsion theory (Flat(C(R)); Flat(C(R))−)have enough injectives in C(R). As an application, we prove that for each atcomplex F and each complex Y of R-modules, Exti (F,X)= 0, whenever Ris n-perfect and i > n.

Let $R$ be a prime ring, $I$ be any nonzero ideal of $R$ and $f:Irightarrow R$ be an additivemap. Then skew-Engel condition $langle... langle langle$$f(x),x^{n_1} rangle,x^{n_2} rangle ,...,x^{n_k} rangle=0$ implies that $f (x)=0$ $forall,xin I$ provided $2neq$ char $(R)>n_1+n_2+...+n_k, $ where $n_1,n_2,...,n_k$ are natural numbers. This extends some existing results. In the end, we also gener...

In the title compound, C(26)H(22)F(2)O(2), the cyclo-propane ring makes dihedral angles of 47.6 (2), 51.3 (2) and 63.9 (2)° with the 2-fluoro-substituted phenyl ring, the unsubstituted phenyl ring and the 4-fluoro-substituted phenyl ring, respectively. There is a short C-H⋯F contact in the molecule. In the crystal, weak C-H⋯F hydrogen bonds lead to chains of mol-ecules extending along the b-axi...

Abstract. In this paper, we define the notion of C*-affine maps in the unital *-rings and we investigate the C*-extreme points of the graph and epigraph of such maps. We show that for a C*-convex map f on a unital *-ring R satisfying the positive square root axiom with an additional condition, the graph of f is a C*-face of the epigraph of f. Moreover, we prove som...

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.

The title compound, C(15)H(13)F(3)N(2)O, contains one benzene ring, one cyclo-hexane ring and a pyridazine ring. The dihedral angle formed by the pyridazine ring with the benzene ring is 61.5 (2)°. The crystal structure is stabilized by two inter-molecular hydrogen bonds (N-H⋯O and C-H⋯F). The cyclohexane ring adopts a screw-boat conformation. The CF(3) group is disordered over two positions; t...

let $f$ be a finite extension of $bbb q$‎, ‎${bbb q}_p$ or a global‎ ‎field of positive characteristic‎, ‎and let $e/f$ be a galois extension‎. ‎we study the realization fields of‎ ‎finite subgroups $g$ of $gl_n(e)$ stable under the natural‎ ‎operation of the galois group of $e/f$‎. ‎though for sufficiently large $n$ and a fixed‎ algebraic number field $f$ every its finite extension $e$ is‎ ‎re...

let $r$ be an arbitrary ring with identity and $m$ a right $r$-module with $s=$ end$_r(m)$. the module $m$ is called {it rickart} if for any $fin s$, $r_m(f)=se$ for some $e^2=ein s$. we prove that some results of principally projective rings and baer modules can be extended to rickart modules for this general settings.

‎for any archimedean$f$-ring $a$ with unit in whichbreak$awedge‎‎(1-a)leq 0$ for all $ain a$‎, ‎the following are shown to be‎‎equivalent‎:‎‎1‎. ‎$a$ is isomorphic to the $l$-ring ${mathfrak z}l$ of all‎‎integer-valued continuous functions on some frame $l$‎. 2‎. ‎$a$ is a homomorphic image of the $l$-ring $c_{bbb z}(x)$‎‎of all integer-valued continuous functions‎, ‎in the usual sense‎,‎on som...