Right serial skew Laurent series rings

Let [Formula: see text] be a ring and [Formula: see text] its automorphism. It is proved that the skew Laurent series ring [Formula: see text] is a right serial ring if and only if [Formula: see text] is a right serial right Artinian ring.

The results of research on tillage rollers with an economic assessment of the effectiveness of their use

The study confirmed the set scientific hypothesis that to improve seed germination and uniformity of germination of cultivated agricultural crops and increase their yields possible while performing operations field preparation, sowing, formation of the desired profile, density and structure of the soil over the sown seeds on the basis of application of innovative methods and means of mechanization of surface tillage. As a result of experimental studies, the design parameters of the developed innovative agricultural tools have been optimized. When combined, they provide the required fine-grained and optimally compacted soil layer, in which the seeds of agricultural crops are located. As recommendations to agricultural production, the actual value of the specific mass of the developed rollers is established, taking into account ballasting loads per 1 m of the width of the grip, the value of which is within 95...160 kg, depending on the shape and design of the working elements, as well as on the properties of the processed medium. In the course of evaluating the metal capacity of innovative agricultural tools and serial ring-spur roller, a difference of 64% per unit of widthof the grip was revealed. Defined increase of productivity when using the developed means of mechanization of surface tillage for spring barley to 14% and 16 %, spring wheat by 11 % and 14 %, winter wheat up to 7% and 9% respectively compared with serial ring-spur roller and disc rollers seeder. Based on the economic assessment of the compared seeding technologies, the annual economic effect of the introduction of innovative tillage rollers was established, which amounted to 64…69 $ by 1 ha of spring barley crops.

Rings all of whose prime serial modules are serial

It is well known that the concept of left serial ring is a Morita invariant property and a theorem due to Nakayama and Skornyakov states that “for a ring [Formula: see text], all left [Formula: see text]-modules are serial if and only if [Formula: see text] is an Artinian serial ring”. Most recently the notions of “prime uniserial modules” and “prime serial modules” have been introduced and studied by Behboodi and Fazelpour in [Prime uniserial modules and rings, submitted; Noetherian rings whose modules are prime serial, Algebras and Represent. Theory 19(4) (2016) 11 pp]. An [Formula: see text]-module [Formula: see text] is called prime uniserial ( [Formula: see text]-uniserial) if its prime submodules are linearly ordered with respect to inclusion, and an [Formula: see text]-module [Formula: see text] is called prime serial ( [Formula: see text]-serial) if [Formula: see text] is a direct sum of [Formula: see text]-uniserial modules. In this paper, it is shown that the [Formula: see text]-serial property is a Morita invariant property. Also, we study what happens if, in the above Nakayama–Skornyakov Theorem, instead of considering rings for which all modules are serial, we consider rings for which every [Formula: see text]-serial module is serial. Let [Formula: see text] be Morita equivalent to a commutative ring [Formula: see text]. It is shown that every [Formula: see text]-uniserial left [Formula: see text]-module is uniserial if and only if [Formula: see text] is a zero-dimensional arithmetic ring with [Formula: see text] T-nilpotent. Moreover, if [Formula: see text] is Noetherian, then every [Formula: see text]-serial left [Formula: see text]-module is serial if and only if [Formula: see text] is serial ring with dim[Formula: see text].

On generalizations of injective modules

As a proper generalization of injective modules in term of supplements, we say that a module M has the property (SE) (respectively, the property (SSE)) if, whenever M ( N, M has a supplement that is a direct summand of N (respectively, a strong supplement in N). We show that a ring R is a left and right artinian serial ring with Rad(R)2 = 0 if and only if every left R-module has the property (SSE). We prove that a commutative ring R is an artinian serial ring if and only if every left R-module has the property (SE).

Global dimension of Harada rings and serial rings

Baba [On Harada rings and quasi-Harada rings with left global dimension at most 2. Comm. Algebra28(6) (2000) 2671–2684] proved that every left Harada rings with global dimension at most 2 is a serial ring. In this paper, improving the result, we show that every left Harada ring with global dimension at most 3 is a serial ring. We also prove that if a left Harada ring A of finite global dimension is of type (*) or has homogeneous right socle, then A is serial. Finally, we give an example of a non-serial left Harada ring of finite global dimension.

A CONSTRUCTION OF RIGHT ARTINIAN RIGHT SERIAL RINGS

A ring R is said to be right serial, if it is a direct sum of right ideals which are uniserial. A ring that is right serial need not be left serial. Right artinian, right serial ring naturally arise in the study of artinian rings satisfying certain conditions. For example, if an artinian ring R is such that all finitely generated indecomposable right R-modules are uniform or all finitely generated indecomposable left R-modules are local, then R is right serial. Such rings have been studied by many authors including Ivanov, Singh and Bleehed, and Tachikawa. In this paper, a universal construction of a class of indecomposable, non-local, basic, right artinian, right serial rings is given. The construction depends on a right artinian, right serial ring generating system X, which gives rise to a tensor ring T(L). It is proved that any basic right artinian, right serial ring is a homomorphic image of one such T(L).

Rad-⊕-Supplemented Modules

In this paper we provide various properties of Rad-⊕-supplemented modules. In particular, we prove that a projective module M is Rad- ⊕-supplemented if and only if M is ⊕-supplemented, and then we show that a commutative ring R is an artinian serial ring if and only if every left R-module is Rad-⊕-supplemented. Moreover, every left R-module has the property (P*) if and only if R is an artinian serial ring and J2 = 0, where J is the Jacobson radical of R. Finally, we show that every Rad-supplemented module is Rad-⊕-supplemented over dedekind domains.