AN EXAMPLE OF A CHAIN PRIME RING WITH NILPOTENT ELEMENTS

<abstract><p>An associative Artinian ring with an identity is a chain ring if its lattice of left (right) ideals forms a unique chain. In this article, we first prove that for every chain ring, there exists a certain finite commutative chain subring which characterizes it. Using this fact, we classify chain rings with invariants $ p, n, r, k, k', m $ up to isomorphism by finite commutative chain rings ($ k' = 1 $). Thus the classification of chain rings is reduced to that of finite commutative chain rings.</p></abstract>

Locally Archimedian right-ordered groups and locally invariant valuation rings

Let [Formula: see text] be a prime ring, [Formula: see text] a noncentral Lie ideal of [Formula: see text] and [Formula: see text] the extended centroid of [Formula: see text], where [Formula: see text] is the Utumi quotient ring of [Formula: see text]. Suppose that [Formula: see text] are two nonzero generalized derivations of [Formula: see text]. Recently, [Dhara et al., Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Comm. Algebra 48(1) (2020) 154–167] studied the case [Formula: see text] for all [Formula: see text], where [Formula: see text] is a derivation (which is independent of [Formula: see text]) and [Formula: see text] fixed positive integers and obtained the structure of [Formula: see text] and the ring [Formula: see text]. [Liu, An Engel condition with two generalized derivations on Lie ideals of prime rings, J. Algebra and Its Applications, 22(5) (2023) 2350115] studied a similar identity replacing derivation [Formula: see text] with another generalized derivation [Formula: see text]. Recently, [Pandey, Pair of generalized derivations on Lie ideals in prime rings, Asian-European J. Math. 16(1) (2022) 2350002] studied an identity considering composition of mapping, that is [Formula: see text] for all [Formula: see text]. In this paper, we consider the situation containing composition of mappings as well as Engel condition, that is, [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed integers and then obtain the nature of mappings [Formula: see text], [Formula: see text] and the ring [Formula: see text].

Let K[G] denote the group ring of G over the field K. In this note we characterize those group rings in which all left ideals are right ideals. Let R be a ring. We say that R is l.i.r.i. if every left ideal is a right ideal. A ring is l.a.r.i. if every left annihilator is a right ideal. Our notation follows that of [2]. The main results are THEOREM I. Let K be a field and let G be a nonabelian periodic group. Then if K[G] is l.a.r.i. one of the following occurs (i) Char K = 0 and G is a Hamiltonian group such that for each odd exponent, n, of G the quaternion algebra over the field K(4), where 4 is a primitive nth root of unity, is a division ring. (ii) Char K = 2 and K does not contain any primitive cube root of unity. Moreover G = Q x A, where Q is the quaternion group of order 8 and A is abelian in which each element has odd order and if n is an exponent for A, the least integer m > 1 satisfying 2m -1 (mod n) is odd. Conversely if K[G] satisfies either (i) or (ii), then K[G] is l.i.r.i. and, in particular, it is l.a.r.i. Observe that if char K > 2 and G is periodic, then K[GI is l.a.r.i. if and only if G is abelian. THEOREM II. Let K[G] denote the group ring over a nonabelian group. Then the following are equivalent (i) K[G] is l.i.r.i. (ii) G is locally finite and if a,f8 E K[GI with a,8 = 0, then /3a = 0. (iii) G is locally finite and K[G] is l.a.r.i. If we combine the above theorems we get necessary and sufficient conditions for K[GI to be l.i.r.i. By using the antiautomorphism of K[G] given by ,xeG k.x H* IeG kxxwe see that K[G] is l.i.r.i. (l.a.r.i.) if and only if K[G] is r.i.l.i. (r.a.l.i.). Received by the editors February 2, 1978. AMS (MOS) subject classifications (1970). Primary 16A26.

In [10], it is defined that a right (or left) ideal I of a ring R is very large if the cardinality of R/I is finite. It is also proven in [10, Theorem 3.4] that if R is a prime ring with 1 such that its characteristic is zero, then R is a right order in a simple ring with the minimum condition on one sided ideals if every large right ideal of R is very large. In the present note, we shall prove that if R is a prime ring with 1 such that its characteristic is zero and R is also a compact topological ring, then R is a right and left order in a simple ring with the minimum condition on one sided ideals, which is also a non-discrete locally compact topological ring if and only if every large right ideal of R is open. In particular, if R is an integral domain with 1 (not necessarily commutative) such that its characteristic is zero, then R is openly embeddable [13, p. 58] in a locally compact (topological) division ring if and only if every large right ideal of R is open. Following S. Warner [13], we shall say R is openly embeddable in a quotient ring Q(R) if there is a topology on Q(R) which is compatible with its structure, which induces on R its given topology and for which R is an open subset.

Let B be any nite commutative ring with identity. In this case, B(X; 1 p kZ0) B(X; 1 p 2Z0) B(X; 1 p Z0), where p is a prime number and k 1, is the descending chain of commutative semigroup rings. All these semigroup rings are containing the polynomial ring B(X;Z0). In this paper, we introduced a construction technique of cyclic codes through the semi- group ring B(X; 1 p kZ0) instead of a polynomial ring.

A right cone of a group G is a submonoid H of G so that for a, b ∈ H either aH ⊆ bH or bH ⊂ aH and G = {ab −1 | a, b ∈ H}. Valuation rings, right chain rings, the cones of right ordered groups provide examples. It is proved, see Theorem 17, that a semi-invariant right cone H with d.c.c. for prime ideals satisfies Ha ⊆ aH for all a ∈ H, that is H is right invariant. Essential is the following Theorem 9: Let H be a locally invariant right cone in G with d.c.c. for prime ideals, and let I ≠ H be an ideal in H. Then P l (I) ⊆ P r (I) for the associated left and right prime ideals of I.

Let R be a topological (Hausdorff) ring such that for each a ∊ R, aR and Ra are closed subsets of R. We will prove that if the set of non - trivial right (left) zero divisors of R is a non-empty set and the set of all right (left) zero divisors of R is a compact subset of R, then R is a compact ring. This theorem has an interesting corollary. Namely, if R is a discrete ring with a finite number of non - trivial left or right zero divisors then R is a finite ring (Refer [1]).

Left dihedral codes over finite chain rings

Let A: be a field and R the ^-algebra generated by x and j with the single defining relation x"=Q.Using free ring techniques we prove that the set of left zero-divisors of R is Rx.There is a unique involution fixing x, y and this makes R into a prime ring with involution whose symmetric zero-divisors are nilpotent (answering a question by W. S. Martindale).This example also provides us with a subfunctor of the identity whose value is a onesided ideal (answering a question by R. Baer).

Let R be a ring. An element p of R is a prime element if pR = Rp is a prime ideal of R. A prime ring R is said to be a Unique Factorisation Ring if every non-zero prime ideal contains a prime element. This paper develops the basic theory of U.F.R.s. We show that every polynomial extension in central indeterminates of a U.F.R. is a U.F.R. We consider in more detail the case when a U.F.R. is either Noetherian or satisfies a polynomial identity. In particular we show that such a ring R is a maximal order, that every height-1 prime ideal of R has a classical localisation in which every two-sided ideal is principal, and that R is the intersection of a left and right Noetherian ring and a simple ring.

An operationally simple and time-efficient approach has been developed for the synthesis of racemic N-substituted 3-(2-aryl-2-oxoethyl)-3-hydroxyindolin-2-ones by a piperidine-catalysed aldol reaction between aryl methyl ketones and N-alkylisatins. These aldol products were used successfully as strategic intermediates for the preparation of N-substituted (E)-3-(2-hetaryl-2-oxoethylidene)indolin-2-ones by a stereoselective dehydration reaction under acidic conditions. The products have all been fully characterized by 1H and 13C NMR spectroscopy, by mass spectrometry and, for a representative selection, by crystal structure analysis. In each of (RS)-1-benzyl-3-hydroxy-3-[2-(4-methoxyphenyl)-2-oxoethyl]indolin-2-one, C24H21NO4, (Ic), and (RS)-1-benzyl-3-{2-[4-(dimethylamino)phenyl]-2-oxoethyl}-3-hydroxyindolin-2-one, C25H24N2O3, (Id), inversion-related pairs of molecules are linked by O-H...O hydrogen bonds to form R22(10) rings, which are further linked into chains of rings by a combination of C-H...O and C-H...π(arene) hydrogen bonds in (Ic) and by C-H...π(arene) hydrogen bonds in (Id). The molecules of (RS)-1-benzyl-3-hydroxy-3-[2-oxo-2-(pyridin-4-yl)ethyl]indolin-2-one, C22H18N2O3, (Ie), are linked into a three-dimensional framework structure by a combination of O-H...N, C-H...O and C-H...π(arene) hydrogen bonds. (RS)-3-[2-(Benzo[d][1,3]dioxol-5-yl)-2-oxoethyl]-1-benzyl-3-hydroxyindolin-2-one, C24H19NO5, (If), crystallizes with Z' = 2 in the space group P-1 and the molecules are linked into complex sheets by a combination of O-H...O, C-H...O and C-H...π(arene) hydrogen bonds. In each of (E)-1-benzyl-3-[2-(4-fluorophenyl)-2-oxoethylidene]indolin-2-one, C23H16FNO2, (IIa), and (E)-1-benzyl-3-[2-oxo-2-(thiophen-2-yl)ethylidene]indolin-2-one, C21H15NO2S, (IIg), the molecules are linked into simple chains by a single C-H...O hydrogen bond, while those of (E)-1-benzyl-3-[2-oxo-2-(pyridin-4-yl)ethylidene]indolin-2-one, C22H16N2O2, (IIe), are linked by three C-H...O hydrogen bonds to form sheets which are further linked into a three-dimensional structure by C-H...π(arene) hydrogen bonds. There are no hydrogen bonds in the structures of either (E)-1-benzyl-3-[2-(4-methoxyphenyl)-2-oxoethylidene]indolin-2-one, C24H19NO3, (IIc), or (E)-1-benzyl-5-chloro-3-[2-(4-chlorophenyl)-2-oxoethylidene]indolin-2-one, C23H15Cl2NO2, (IIh), but the molecules of (IIh) are linked into chains of π-stacked dimers by a combination of C-Cl...π(arene) and aromatic π-π stacking interactions.

Twelve 4-(4-meth-oxy-phen-yl)piperazin-1-ium salts containing organic anions have been prepared and structurally characterized. The monohydrated benzoate, 4-fluoro-benzoate, 4-chloro-benzoate and 4-bromo-benzoate salts, C11H17N2O+·C7H5O2 -·H2O (I), C11H17N2O+·C7H4FO2 -·H2O (II), C11H17N2O+·C7H4ClO2 -·H2O (III), and C11H17N2O+·C7H4BrO2 -·H2O (IV), respectively, are isomorphous and all exhibit disorder in the 4-meth-oxy-phenyl unit: the components are linked by N-H⋯O and O-H⋯O hydrogen bond to form chains of rings. The unsolvated 2-hy-droxy-benzoate, pyridine-3-carboxyl-ate and 2-hy-droxy-3,5-di-nitro-benzoate salts, C11H17N2O+·C7H5O3 - (V), C11H17N2O+·C6H4NO2 - (VI) and C11H17N2O+·C7H3N2O7 - (VII), respectively, are all fully ordered: the components of (V) are linked by multiple N-H⋯O hydrogen bonds to form a chain of rings; those of (VI) are linked into a three-dimensional framework by a combination of N-H⋯O, C-H⋯O and C-H⋯N hydrogen bonds and those of (VII), where the anion has a structure reminiscent of the picrate anion, are linked into a three-dimensional array by N-H⋯O and C-H⋯O hydrogen bonds. The hydrogensuccinate and hydrogenfumarate salts, C11H17N2O+·C4H5O4 - (VIII) and C11H17N2O+·C4H3O3 - (IX), respectively, are isomorphous, and both exhibit disorder in the anionic component: N-H⋯O and O-H⋯O hydrogen bonds link the ions into sheets, which are further linked by C-H⋯π(arene) inter-actions. The anion of the hydrogenmaleate salt, C11H17N2O+·C4H3O3 - (X), contains a very short and nearly symmetrical O⋯H⋯O hydrogen bond, and N-H⋯O hydrogen bonds link the anions into chains of rings. The ions in the tri-chloro-acetate salt, C11H17N2O+·C2Cl3O2 - (XI), are linked into simple chains by N-H⋯O hydrogen bonds. In the hydrated chloranilate salt, 2C11H17N2O+·C6Cl2O4 2-·2H2O (XII), which crystallizes as a non-merohedral twin, the anion lies across a centre of inversion in space group P21/n, and a combination of N-H⋯O and O-H⋯O hydrogen bonds generates complex sheets. Comparisons are made with the structures of some related compounds.

New triarylpyrazole derivatives, substituted with basic side chain, polar group or heterocyclic ring were designed and synthesized. Nineteen compounds were tested for their in vivo uterotrophic activity. Animals treated with compounds 9, 15a, 15c, 19a, 19f, 20a, and 21b showed pronounced increase in uterine weight. It is worth mentioning that compound 19f was more active than estradiol as an estrogenic compound as evidenced by the increased dry uterine weight. Additionally, compound 19a produced 128.25% increase in dry uterine weight relative to estradiol. On the other hand, eight compounds were screened for their in vivo antiuterotrophic activity. Animals treated with compounds 5e, 5f, 8, and 13b showed pronounced decrease in uterine weight. The highest antiuterotrophic effect was observed in mice receiving compound 13b, which was as potent as the positive control, tamoxifen as evident by the decrease in the uterine wet weight.

Detection of 58 fentanyl analogs using ARK fentanyl II and Immunalysis fentanyl immunoassays