(Jordan) derivation on amalgamated duplication of a ring along an ideal

Let A be a ring and I be an ideal of A. The amalgamated duplication of A along I is the subring of A × A defined by $A\bowtie I := {(a, a + i) |a ∈ A, i ∈ I}$. In this paper, we characterize $A\bowtie I$ over which any (resp. minimal) prime ideal is invariant under any derivation provided that A is semiprime. When A is noncommutative prime, then $A\bowtie I$ is noncommutative semiprime (but not prime except if I = (0)). In this case, we prove that any map of $A\bowtie I$ which is both Jordan and Jordan triple derivation is a derivation.

Birecognition of prime graphs, and minimal prime graphs

Given a graph [Formula: see text], a subset [Formula: see text] of [Formula: see text] is a module of [Formula: see text] if for each [Formula: see text], [Formula: see text] is adjacent to all the elements of [Formula: see text] or to none of them. For instance, [Formula: see text], [Formula: see text] and [Formula: see text] ([Formula: see text]) are the trivial modules of [Formula: see text]. A graph [Formula: see text] is prime if [Formula: see text] and all its modules are trivial. Given a prime graph [Formula: see text], consider [Formula: see text] such that [Formula: see text] is prime. Given a graph [Formula: see text] such that [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] are [Formula: see text]-similar if for each [Formula: see text], [Formula: see text] and [Formula: see text] are both prime or not. The graph [Formula: see text] is said to be [Formula: see text]-birecognizable if every graph, [Formula: see text]-similar to [Formula: see text], is prime. We study the graphs [Formula: see text] that are not [Formula: see text]-birecognizable, where [Formula: see text] such that [Formula: see text] is prime, by using the following notion of a minimal prime graph. Given a prime graph [Formula: see text], consider [Formula: see text] such that [Formula: see text] is prime. Given [Formula: see text], [Formula: see text] is [Formula: see text]-minimal if for each [Formula: see text] such that [Formula: see text], [Formula: see text] is not prime.

Krull modules and completely integrally closed modules

Let [Formula: see text] be a torsion-free module over an integral domain [Formula: see text] with quotient field [Formula: see text]. We define a concept of completely integrally closed modules in order to study Krull modules. It is shown that a Krull module [Formula: see text] is a [Formula: see text]-multiplication module if and only if [Formula: see text] is a maximal [Formula: see text]-submodule and [Formula: see text] for every minimal prime ideal [Formula: see text] of [Formula: see text]. If [Formula: see text] is a finitely generated Krull module, then [Formula: see text] is a Krull module and [Formula: see text]-multiplication module. It is also shown that the following three conditions are equivalent: [Formula: see text] is completely integrally closed, [Formula: see text] is completely integrally closed, and [Formula: see text] is completely integrally closed.

OPTIMIZATION OF STAGES OF PRE-SPAWNING AND SPAWNING PERIODS OF TILAPIA IN COMMERCIAL GROWING

The article describes the process of commercial fish production in modern aquaculture, which requires high technologies and significant input. Optimization of technological processes while breeding fish is a necessary condition for the development of the industry. It is difficult and virtually impossible to develop a universal production pattern, this is why it is reasonable to split the whole process into several optimization tasks corresponding to each of the stages of the technological cycle. It is proposed to regulate such indicators as the level of feeding, pH value, age of producers as control actions at the stages of pre-spawning and spawning periods. The possibility to use the mentioned parameters for stage optimization, as well as for selecting the best values for each of the factors was assessed. The optimization target is to define the optimal amount of viable eggs, which would correspond to the minimal prime cost. It has been demonstrated that the optimal feeding level for the producers equals to 4 – 4.5 per cent of the fish body weight. The increase in the feeding level influences the fertility of producers insufficiently, inflating the costs. It has been found out that 12 – 24 months old producers have the best reproduction indices. The preferred quality and the vital capacity of spawn received from the producers within this group have also been registered. Another factor which may be used in order to control the process is the pH of water in pools. Regardless of the fact that tilapia producers are relatively undemanding to the habitat conditions and resistant to pH values, it is possible to single out the optimal range of this factor. It corresponds to pH values between 6 and 7.5. This is when the best rates of eggs fertilization and survival are observed.

On $n$-Absorbing Ideals in a Lattice

Let L be a lattice, and let n be a positive integer. In this article, we introduce n-absorbing ideals in L. We give some properties of such ideals. We show that every n-absorbing ideal I of L has at most n minimal prime ideals. Also, we give some properties of 2-absorbing and weakly 2-absorbing ideals in L. In particular we show that in every non-zero distributive lattice L, 2-absorbing and weakly 2-absorbing ideals are equivalent.

Maximal prime homomorphic images of mod-p Iwasawa algebras

Let k be a finite field of characteristic p, and G a compact p-adic analytic group. Write kG for the completed group ring of G over k. In this paper, we describe the structure of the ring kG/P, where P is a minimal prime ideal of kG. We give an explicit isomorphism between kG/P and a matrix ring with coefficients in the ring ${(k'G')_\alpha }$ , where $k'/k$ is a finite field extension, $G'$ is a large subquotient of G with no finite normal subgroups, and (–) α is a “twisting” operation that preserves many desirable properties of the ring structure. We demonstrate the usefulness of this isomorphism by studying the correspondence induced between certain ideals of kG and those of ${(k'G')_\alpha }$ , and showing that this preserves many useful “group-theoretic” properties of ideals, in particular almost-faithfulness and control by a closed normal subgroup.