Intermediate ideals in matrix near-rings

Meldrum, J. D. P.; Meyer, J. H.

doi:10.1080/00927879608825658

Cited by 10 publications

(6 citation statements)

References 6 publications

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“…Matrix nearrings over arbitrary nearrings were introduced in Meldrum & Van der Walt [11], where several results about the correspondence between the two-sided ideals in the base nearring N and those in the matrix nearring M n (N ) were proved. Later, remarkable developments in matrix nearrings over arbitrary nearrings were due to Meldrum and Meyer [12], Meyer [13,15]. 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 .…”

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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Section: Introductionmentioning

confidence: 99%

Partial Order in Matrix Nearrings

Sahoo

Meyer

Panackal

et al. 2022

Bull. Iran. Math. Soc.

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“…(see [1] and [4]), several questions were raised in connection with these ideals. A fact which followed immediately was that the J 2 -radical of a matrix nearring can never be intermediate-for any (zerosymmetric) nearring R we have J 2 M n (R) Á ≥ J 2 (R)…”

Section: Introduction Soon After the Discovery Of Intermediate Idealmentioning

confidence: 99%

“…Á Ł (see [7,Theorem 4.4]). Because this relation does not hold for the J 0 -radical in general (see [2]), the question was raised in [1] whether the J 0 -radical of a matrix nearring can be intermediate. The object of this note is to provide an example of a finite zerosymmetric abelian nearring R for which J 0 M n (R) Á is an intermediate ideal.…”

Section: Introduction Soon After the Discovery Of Intermediate Idealmentioning

confidence: 99%

“…It easily follows that A + Â A Ł , and several examples exist (see [6], [2], [1] and [4]) to show that A + ² Â ≥ A Ł is possible. It is also possible that ideals of M n (R) can be properly situated between A + and A Ł (see [1] and [4]). Any ideal I of M n (R) with the property…”

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confidence: 96%

“…In [1] it is shown that an intermediate ideal can never be of the form B + or B Ł for any ideal B of R. In the next section we construct a nearring R for which J 0 M n (R) Á is intermediate.…”

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confidence: 99%

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