On Finite Goldie Dimension of Mn(N)-Group Nn

Satyanarayana, B.; Kuncham, Syam Prasad

doi:10.1007/1-4020-3391-5_17

Cited by 4 publications

(5 citation statements)

References 5 publications

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“…The following Lemma 3.1 and Lemma 3.3 from [9] are useful, and so, for completeness we brief the proofs. Proof Let I be weakly prime in N N .…”

Section: Weakly Prime Ideals Of M N (N)-group N Nmentioning

confidence: 99%

“…Further, Juglal and Groenewald [8] studied the class of strongly prime nearring modules and shown that it forms a -special class. In Bhavanari and Kuncham [9], the relation between the ideals of the N-group N and the ideals of M n (N)-group N n has been studied. Badawi and Darani [10] studied weakly 2-absorbing ideals as a generalization of prime ideals in commutative rings.…”

Section: Introductionmentioning

confidence: 99%

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Generalization of prime ideals in $$M_n(N)$$-group $$N^{n}$$

Tapatee

Kedukodi

Juglal

et al. 2021

Rend. Circ. Mat. Palermo, II. Ser

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The notion of a matrix nearring over an arbitrary nearring was introduced by (Meldrum and Walt Arch. Math. 47(4): 312–319, 1986). In this paper, we define the notions such as weakly $$\tau$$ τ -prime $$(\tau =0,c,3,e)$$ ( τ = 0 , c , 3 , e ) ideals of an N-group G, which are the generalization of the classes of $$\tau$$ τ -prime ideals of G, and provide suitable examples to distinguish between the two classes. We extend the concept to obtain the one-one correspondence between weakly $$\tau$$ τ -prime ideals $$(\tau =0,c,3,e)$$ ( τ = 0 , c , 3 , e ) of N-group (over itself) and those of $$M_n(N)$$ M n ( N ) -group $$N^{n}$$ N n , where $$M_n(N)$$ M n ( N ) is the matrix nearring over the nearring N. Further, we prove the correspondence between weakly 2-absorbing ideals of these classes.

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“…The following Lemma 3.1 and Lemma 3.3 from [9] are useful, and so, for completeness we brief the proofs. Proof Let I be weakly prime in N N .…”

Section: Weakly Prime Ideals Of M N (N)-group N Nmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalization of prime ideals in $$M_n(N)$$-group $$N^{n}$$

Tapatee

Kedukodi

Juglal

et al. 2021

Rend. Circ. Mat. Palermo, II. Ser

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“…The notion of finite Goldie dimension (denoted by FGD) of a module was defined by Goldie [14] wherein, the key notions for the study of FGD are essential submodules, uniform sub-modules and complement of a submodule (see, [4,5]). The dualization of this concept namely, finite spanning dimension (denoted by FSD) in modules over rings was defined by Fleury [13] with the notions such as superfluous submodules, hollow submodules and supplements.…”

Section: Introductionmentioning

confidence: 99%

Superfluous ideals of N-groups

Rajani

Tapatee

Harikrishnan

et al. 2023

Rend. Circ. Mat. Palermo, II. Ser

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We consider a right nearring N and a module over N (known as, N-group). For an arbitrary ideal (or N-subgroup) $$\varOmega $$ Ω of an N-group G, we define the notions $$\varOmega $$ Ω -superfluous, strictly $$\varOmega $$ Ω -superfluous, g-superfluous ideals of G. We give suitable examples to distinguish between these classes and the existing classes studied in Bhavanari (Proc Japan Acad 61-A:23–25, 1985; Indian J Pure Appl Math 22:633–636, 1991; J Austral Math Soc 57:170–178, 1994), and prove some properties. For a zero-symmetric nearring with 1, we consider a module over a matrix nearring and obtain one-one correspondence between the superfluous ideals of an N-group (over itself) and those of $$M_{n}(N)$$ M n ( N ) -group $$N^{n}$$ N n , where $$M_{n}(N)$$ M n ( N ) is the matrix nearring over N. Furthermore, we define a graph of superfluous ideals of a nearring and prove some properties with necessary examples.

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“…Meldrum and Meyer [12] have shown the existence of an arbitrary large lattice of ideals in the matrix nearring corresponding to an ideal in the base nearring N . More concepts in the ideal theory of matrix nearrings are due to [1,3,4]. Van der Walt [20,21] studied the relationship between the modules over a nearring N , and those in modules over a matrix nearring M n (N ).…”

Section: Introductionmentioning

confidence: 99%

“…In Sect. 4, the convex ideal in the matrix nearring corresponding to a convex ideal in the base nearring N is introduced. Further, it is proved that if I is a convex ideal of N , then the corresponding ideal I * is convex in M n (N ); conversely, it is shown that for a convex ideal I in M n (N ), I * is convex in the base nearring N .…”

Section: Introductionmentioning

confidence: 99%

Partial Order in Matrix Nearrings

Sahoo

Meyer

Panackal

et al. 2022

Bull. Iran. Math. Soc.

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Let N be a zero-symmetric (right) nearring with identity. We introduce a partial order in the matrix nearring corresponding to the partial order (defined by Pilz in Near-rings: the theory and its applications, North Holland, Amsterdam, 1983) in N. A positive cone in a matrix nearring is defined and a characterization theorem is obtained. For a convex ideal I in N, we prove that the corresponding ideal $${I^*}$$ I ∗ is convex in $$M_n(N)$$ M n ( N ) , and conversely, if I is convex in $$M_n(N)$$ M n ( N ) , then $${I_{*}}$$ I ∗ is convex in N. Consequently, we establish an order-preserving isomorphism between the p.o. quotient matrix nearrings $$M_n(N)/{I^*}$$ M n ( N ) / I ∗ and $$M_n(N')/{(I')^*}$$ M n ( N ′ ) / ( I ′ ) ∗ where I and $$I'$$ I ′ are the convex ideals of p.o. nearrings N and $$N'$$ N ′ , respectively. Finally, we prove some properties of Archimedean ordering in matrix nearrings corresponding to those in nearrings.

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On Finite Goldie Dimension of Mn(N)-Group Nn

Cited by 4 publications

References 5 publications

Generalization of prime ideals in $$M_n(N)$$-group $$N^{n}$$

Generalization of prime ideals in $$M_n(N)$$-group $$N^{n}$$

Superfluous ideals of N-groups

Partial Order in Matrix Nearrings

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