Radicals of semirings

2019 ◽  
Vol 13 (07) ◽  
pp. 2050138
Author(s):  
Ram Parkash Sharma ◽  
Richa Sharma ◽  
Madhu
Keyword(s):  
Invariant Property ◽  
Ideal Formula ◽  
Commutative Semiring ◽  
Morita Invariant

It is shown that the classes [Formula: see text] and [Formula: see text] of semirings are radical classes, where [Formula: see text] is the class of subtractive-simple right [Formula: see text]-semimodules and [Formula: see text] is the class of right [Formula: see text]-semimodules isomorphic to [Formula: see text] for some maximal-subtractive right ideal [Formula: see text] of [Formula: see text]. We define the lower Jacobson Bourne radical [Formula: see text] and upper Jacobson Bourne radical [Formula: see text] of [Formula: see text]. For a semiring [Formula: see text], [Formula: see text] holds, where [Formula: see text] is the Jacobson Bourne radical of [Formula: see text]. The radical [Formula: see text] and also coincides with [Formula: see text], if we restrict the class [Formula: see text] to additively cancellative semimodules[Formula: see text] The upper radical [Formula: see text] and [Formula: see text][Formula: see text], if [Formula: see text] is additively cancellative. Further, [Formula: see text], if [Formula: see text] is a commutative semiring with [Formula: see text] The subtractive-primitiveness and subtractive-semiprimitiveness of [Formula: see text] are closely related to the upper radical [Formula: see text] Finally, we show that [Formula: see text]-semisimplicity of semirings are Morita invariant property with some restrictions.

scholarly journals Central elements of the Jennings basis and certain Morita invariants

2019 ◽  
Vol 19 (08) ◽  
pp. 2050160 ◽  
Author(s):  
Taro Sakurai
Keyword(s):  
Group Algebra ◽  
Positive Characteristic ◽  
Finite Dimensional Algebra ◽  
Independent Subset ◽  
Dimensional Algebra ◽  
Ideal Formula ◽  
Finite Dimensional ◽  
Linearly Independent ◽  
Morita Invariant ◽  
Central Elements

From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the [Formula: see text]th (right) socle [Formula: see text] of a finite-dimensional algebra [Formula: see text] is a Morita invariant; this is a generalization of important Morita invariants — the center [Formula: see text] and the Reynolds ideal [Formula: see text]. As an example, we also studied [Formula: see text] for the group algebra FG of a finite [Formula: see text]-group [Formula: see text] over a field [Formula: see text] of positive characteristic [Formula: see text]. Such an algebra has a basis along the socle filtration, known as the Jennings basis. We prove certain elements of the Jennings basis are central and hence form a linearly independent subset of [Formula: see text]. In fact, such elements form a basis of [Formula: see text] for every integer [Formula: see text] if [Formula: see text] is powerful. As a corollary we have [Formula: see text] if [Formula: see text] is powerful.

On the Yoneda Ext-Algebras of Semiperfect Algebras

2008 ◽  
Vol 15 (02) ◽  
pp. 207-222 ◽  
Author(s):  
Jiwei He ◽  
Yu Ye
Keyword(s):  
Jacobson Radical ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Yoneda Algebra ◽  
Ideal Formula ◽  
Morita Equivalent ◽  
Morita Invariant ◽  
The Right ◽  
Necessary And Sufficient

It is proved that the Yoneda Ext-algebras of Morita equivalent semiperfect algebras are graded equivalent. The Yoneda Ext-algebras of noetherian semiperfect algebras are studied in detail. Let A be a noetherian semiperfect algebra with Jacobson radical J. We construct a right ideal [Formula: see text] of the Yoneda algebra [Formula: see text], which plays an important role in the discussion of the structure of E(A). An extra grading is introduced to [Formula: see text], by which we give a description of the right ideal of E(A) generated by [Formula: see text], and we give a necessary and sufficient condition for a notherian semiperfect algebra to be higher quasi-Koszul. Finally, it is shown that the quasi-Koszulity of a noetherian semiperfect algebra is a Morita invariant.

Weak Quasi-Armendariz Rings

2011 ◽  
Vol 18 (04) ◽  
pp. 541-552 ◽  
Author(s):  
Muhittin Başer ◽  
Fatma Kaynarca ◽  
Tai Keun Kwak ◽  
Yang Lee
Keyword(s):  
Polynomial Ring ◽  
Semiprime Ring ◽  
Invariant Property ◽  
Morita Invariant ◽  
Armendariz Rings ◽  
The Ideal

In this paper, we introduce and study weak quasi-Armendariz rings which unify the notions of weak Armendariz rings and quasi-Armendariz rings. It is shown that the weak quasi-Armendarizness is a Morita invariant property. For a semiprime ring R, it is shown that R[x]/〈xn〉 is weak quasi-Armendariz, where R[x] is the polynomial ring over R and 〈xn〉 is the ideal of R[x] generated by xn. Various properties of weak quasi-Armendariz rings are also observed.

Rings all of whose prime serial modules are serial

2017 ◽  
Vol 16 (10) ◽  
pp. 1750193
Author(s):  
M. Behboodi ◽  
Z. Fazelpour ◽  
M. R. Vedadi
Keyword(s):  
Commutative Ring ◽  
Noetherian Rings ◽  
Invariant Property ◽  
Serial Ring ◽  
Direct Sum ◽  
Morita Equivalent ◽  
Morita Invariant ◽  
Prime Submodules ◽  
Uniserial Modules

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].

An isotopically invariant property of automorphic Moufang loops

2019 ◽  
Vol 58 (4) ◽  
pp. 458-466
Author(s):  
A. N. Grishkov ◽  
M. N. Rasskazova ◽  
L. L. Sabinina
Keyword(s):  
Invariant Property ◽  
Moufang Loops

scholarly journals Pointwise multiple averages for sublinear functions

2018 ◽  
Vol 40 (6) ◽  
pp. 1594-1618
Author(s):  
SEBASTIÁN DONOSO ◽  
ANDREAS KOUTSOGIANNIS ◽  
WENBO SUN
Keyword(s):  
Large Class ◽  
Explicit Formula ◽  
Pointwise Convergence ◽  
Limit Function ◽  
Invariant Property ◽  
Ergodic Averages ◽  
Order Zero ◽  
Good For ◽  
Sublinear Functions ◽  
Uniquely Ergodic

For any measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T_{1},\ldots ,T_{d})$ with no commutativity assumptions on the transformations $T_{i},$$1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order zero and of Fejér functions, i.e., tempered functions of order zero. We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are, in general, bad for convergence on arbitrary systems, but good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

Congruence relations on almost distributive lattices-II

2019 ◽  
pp. 2150005
Author(s):  
Gezahagne Mulat Addis
Keyword(s):  
Distributive Lattice ◽  
Congruence Relation ◽  
Congruence Class ◽  
Distributive Lattices ◽  
Ideal Formula ◽  
Congruence Relations

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

scholarly journals BOUNDED AND FULLY BOUNDED MODULES

2011 ◽  
Vol 84 (3) ◽  
pp. 433-440
Author(s):  
A. HAGHANY ◽  
M. MAZROOEI ◽  
M. R. VEDADI
Keyword(s):  
Krull Dimension ◽  
Finitely Generated ◽  
Morita Invariant ◽  
Prime Submodule ◽  
Invariant Properties ◽  
Noetherian Modules

AbstractGeneralizing the concept of right bounded rings, a module MR is called bounded if annR(M/N)≤eRR for all N≤eMR. The module MR is called fully bounded if (M/P) is bounded as a module over R/annR(M/P) for any ℒ2-prime submodule P◃MR. Boundedness and right boundedness are Morita invariant properties. Rings with all modules (fully) bounded are characterized, and it is proved that a ring R is right Artinian if and only if RR has Krull dimension, all R-modules are fully bounded and ideals of R are finitely generated as right ideals. For certain fully bounded ℒ2-Noetherian modules MR, it is shown that the Krull dimension of MR is at most equal to the classical Krull dimension of R when both dimensions exist.

AN AUTOREGRESSIVE APPROACH TO SURFACE TEXTURE ANALYSIS

1994 ◽  
Vol 08 (04) ◽  
pp. 845-857 ◽  
Author(s):  
DINESH P. MITAL ◽  
GOH WEE LENG
Keyword(s):  
Aspect Ratio ◽  
Random Field ◽  
Texture Analysis ◽  
Surface Texture ◽  
Field Model ◽  
Invariant Property ◽  
Circular Area ◽  
Pixel Image ◽  
Rotational Invariant ◽  
Surface Texture Analysis

The use of autoregressive models in textual analysis holds great potential. Coupling the technique to a circular neighbourhood set imparts a rotational invariant property to it. This was demonstrated by Kashyap and Khotanzad in their model called the Circular Symmetric Autogressive (CSAR) Random Field model. The short-coming in this very ingenious proposal is that it is set in a background of square pixels and the rotational invariant property of the model fails in cases when the aspect ratio of the pixels are not at unity. This paper proposes a major modification to the CSAR to render the model rotational invariant under all configurations of pixel implementation. It is based on the area segments covered by a circle set in a 3×3 neighbourhood. We call it the Circular Area Autoregressive (CAAR) model. The results obtained from the CAAR showed much better consistency over that of the CSAR when a non-square pixel image was used.

Maximal and prime k-subsemimodules in semimodules over semirings

2016 ◽  
Vol 16 (07) ◽  
pp. 1750130 ◽  
Author(s):  
Song-Chol Han
Keyword(s):  
Commutative Semiring ◽  
Simple Semimodule

For a semimodule over any semiring, the maximal and prime [Formula: see text]-subsemimodules are characterized with the help of the quotient structure. The main results are as follows. A semimodule is [Formula: see text]-congruence-simple iff it is [Formula: see text]-subsemimodule-simple. A nonzero semimodule is [Formula: see text]-simple iff it satisfies condition [Formula: see text]. A proper [Formula: see text]-subsemimodule of a semimodule is a maximal [Formula: see text]-subsemimodule iff the quotient semimodule is [Formula: see text]-simple. A proper [Formula: see text]-subsemimodule is prime in a semimodule iff the quotient semimodule is prime. A nonzero [Formula: see text]-simple semimodule over a commutative semiring is prime. Lemma 6.2 in this paper corrects the statement and proof of Lemma 3.16 in [Yeşilot Hacet. J. Math. Stat. 39 (2010) 305–312].

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