THE LOEWY SERIES OF AN FCP (DISTRIBUTIVE) RING EXTENSION

Normal Bases on Galois Ring Extensions

Symmetry ◽

10.3390/sym10120702 ◽

2018 ◽

Vol 10 (12) ◽

pp. 702

Author(s):

Aixian Zhang ◽

Keqin Feng

Keyword(s):

Signal Processing ◽

Finite Field ◽

Block Cipher ◽

Field Extension ◽

Galois Ring ◽

Normal Basis ◽

Galois Rings ◽

Ring Extension ◽

Galois Fields ◽

Normal Bases

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.

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Development of New Photovoltaic Conjugated Polymers Based on Di(1-benzothieno)[3,2-b:2′,3′-d]pyrrole: Benzene Ring Extension Strategy for Improving Open-Circuit Voltage

Macromolecules ◽

10.1021/acs.macromol.5b01129 ◽

2015 ◽

Vol 48 (15) ◽

pp. 5213-5221 ◽

Cited By ~ 27

Author(s):

In Hwan Jung ◽

Ji-Hoon Kim ◽

So Youn Nam ◽

Changjin Lee ◽

Do-Hoon Hwang ◽

...

Keyword(s):

Conjugated Polymers ◽

Benzene Ring ◽

Open Circuit Voltage ◽

Open Circuit ◽

Ring Extension

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Wakamatsu tilting modules over ring extension

Journal of Algebra and Its Applications ◽

10.1142/s0219498819501986 ◽

2019 ◽

Vol 18 (10) ◽

pp. 1950198

Author(s):

Zhen Zhang ◽

Jiaqun Wei

Keyword(s):

Ring Extension ◽

Tilting Module ◽

Tilting Modules ◽

Wakamatsu Tilting Module ◽

Extension Ring

For a ring [Formula: see text], an extension ring [Formula: see text], and a fixed right [Formula: see text]-module [Formula: see text], we prove the induced left [Formula: see text]-module [Formula: see text] is a Wakamatsu tilting module when [Formula: see text] is a Wakamatsu tilting module.

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Ring Extension of Cyclobutane Derivatives

Handbook of Palladium-Catalyzed Organic Reactions ◽

10.1016/b978-012466615-3/50038-0 ◽

1997 ◽

pp. 177-179

Author(s):

J.-L. Malleron ◽

J.-C. Fiaud ◽

J.-Y. Legros

Keyword(s):

Ring Extension ◽

Cyclobutane Derivatives

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Quotient Rings of a Class of Lattice-Ordered-Rings

Canadian Journal of Mathematics ◽

10.4153/cjm-1973-064-3 ◽

1973 ◽

Vol 25 (3) ◽

pp. 627-645 ◽

Cited By ~ 4

Author(s):

Stuart A. Steinberg

Keyword(s):

Identity Element ◽

Quotient Ring ◽

Ring Extension ◽

Qf Ring ◽

Left Quotient ◽

Quotient Rings

An f-ring R with zero right annihilator is called a qf-ring if its Utumi maximal left quotient ring Q = Q(R) can be made into and f-ring extension of R. F. W. Anderson [2, Theorem 3.1] has characterized unital qf-rings with the following conditions: For each q ∈ Q and for each pair d1, d2 ∈ R+ such that diq ∈ R(i) (d1q)+ Λ (d2q)- = 0, and(ii) d1 Λ d2 = 0 implies (d1q)+ Λ d2 = 0.We remark that this characterization holds even when R does not have an identity element.

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On separable extensions of group rings and quaternion rings

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171278000435 ◽

1978 ◽

Vol 1 (4) ◽

pp. 433-438

Author(s):

George Szeto

Keyword(s):

Commutative Ring ◽

Full Description ◽

Group Rings ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Ring Extension ◽

Quaternion Algebras ◽

Separable Group ◽

Necessary And Sufficient ◽

Separable Extensions

The purposes of the present paper are (1) to give a necessary and sufficient condition for the uniqueness of the separable idempotent for a separable group ring extensionRG(Rmay be a non-commutative ring), and (2) to give a full description of the set of separable idempotents for a quaternion ring extensionRQover a ringR, whereQare the usual quaternionsi,j,kand multiplication and addition are defined as quaternion algebras over a field. We shall show thatRGhas a unique separable idempotent if and only ifGis abelian, that there are more than one separable idempotents for a separable quaternion ringRQ, and thatRQis separable if and only if2is invertible inR.

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An Exact Sequence Associated with a Generalized Crossed Product

Nagoya Mathematical Journal ◽

10.1017/s0027763000015269 ◽

1973 ◽

Vol 49 ◽

pp. 21-51 ◽

Cited By ~ 13

Author(s):

Yôichi Miyashita

Keyword(s):

Exact Sequence ◽

Crossed Product ◽

Group Homomorphism ◽

Ring Extension ◽

Exact Sequences

The purpose of this paper is to generalize the seven terms exact sequence given by Chase, Harrison and Rosenberg [8]. Our work was motivated by Kanzaki [16] and, of course, [8], [9]. The main theorem holds for any generalized crossed product, which is a more general one than that in Kanzaki [16]. In §1, we define a group P(A/B) for any ring extension A/B, and prove some preliminary exact sequences. In §2, we fix a group homomorphism J from a group G to the group of all invertible two-sided B-submodules of A.

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