Epimorphic extensions of non-commutative rings

Storrer, Hans H.

doi:10.1007/bf02566112

Cited by 22 publications

(8 citation statements)

References 17 publications

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“…/ : Z -» 5 , maxepi(i/,5) := K{S) acts like the characteristic of S. Indeed, if S is a division ring then K{S) is the prime field of S. On the other hand K(Q x Z/(2)) = Q x Z/(2), revealing the mixture of characteristics of that ring. The ring K(S) is always central [17,Proposition 1.3] and the structure of all epimorphs of Z is known (in fact of all epimorphs of an arbitrary Dedekind domain). (See [1] and [16] for the case of Z and [3] and [5] for the generalization.)…”

Section: The Characteristic Ringmentioning

confidence: 99%

The characteristic ring and the “best” way to adjoin a one

Burgess

Stewart

1989

J Aust Math Soc A

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For any ring 5 we define and describe its characteristic ring, K(S). It plays the role of the usual characteristic even in rings whose additive structure, (S, +), is complicated. The ring K(S) is an invariant of (S, +) and also reflects certain non-additive properties of S. If R is a left faithful ring without identity element, we show how to use K(R) to embed R in a ring R x with identity. This unital overling of R inherits many ring properties of R; for instance, if R is artinian, noetherian, semiprime Goldie, regular, biregular or a K-ring, so too is R l . In the case of regularity (or generalizations thereof), /?' satisfies a universal property with respect to the adjunction of an identity

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Section: The Characteristic Ringmentioning

confidence: 99%

The characteristic ring and the “best” way to adjoin a one

Burgess

Stewart

1989

J Aust Math Soc A

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“…Let R be an extended semihereditary ring. Then (i) If e is an idempotent of R such that ReR = R, then eRe is an extended semihereditary ring, and eM(R)e is the left flat epimorphic hull of eRe [18], (ii) (M(R)) (n) , the ring o f n X w matrices over M(R), is the left flat epimorphic hull of R (n) , the ring o f « X n matrices over R. PROOF, (i) <=> (ii). Let P be a projective i?-module.…”

Section: Extended Semihereditary Ringsmentioning

confidence: 99%

Torsion theories over semihereditary rings

Evans¹

1986

J Aust Math Soc A

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In this paper the class of rings for which the right flat modules form the torsion-free class of a hereditary torsion theory {3~c, &G) are characterized and their structure investigated. These rings are called extended semihereditary rings. It is shown that the class of regular rings with ring homomorphism is a full co-reflective subcategory of the class of extended semihereditary rings with "flat" homomorphisms. A class of prime torsion theories is introduced which determines the torsion theory (SQ, ^C). The torsion theory (S~G, ^G) is used to find a suitable generalisation of Dedekind Domain.

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“…As an example which shows that the balanced condition for modules have strong influence on the structure of rings we can give a theorem of Ringel [7] and Starrer [8], [9] that commutative noetherian strong QF-1 rings are QF (i.e. quasi-Frobenius).…”

Section: Hiroyuki Tachikawamentioning

confidence: 99%

Commutative perfect 𝑄𝐹-1 rings

Tachikawa¹

1978

Proc. Amer. Math. Soc.

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Abstract. If R is a commutative artinian ring, then it is known that every finitely generated faithful Ä-module is balanced (i.e. has the double centralizer property) if and only if R is a quasi-Frobenius ring. In this note, constructing new nonbalanced modules we prove that the assumption on R to be artinian can be replaced by the weaker condition that R is perfect.As an example which shows that the balanced condition for modules have strong influence on the structure of rings we can give a theorem of Ringel [7] and Starrer [8], [9] that commutative noetherian strong QF-1 rings are QF (i.e. quasi-Frobenius). This is a generalization of a theorem of Camillo [1] and Dickson and Fuller [3] for commutative artinian rings, which is also a generalization of Floyd's result [4] for commutative algebras over an algebraically closed field. Here a module M over a ring R with identity 1 is called balanced if the canonical ring homomorphism of R into the double centralizer of M is surjective, and following R. M. Thrall [10] R is said to be QF-1 (resp. strong QF-1) if every finitely generated faithful (resp. every faithful) Ä-module is balanced.In connection with commutative QF-1 rings V. P. Camillo [2] studied rings with the principal extension property (i.e. every module homomorphism from a principal ideal into R can be extended to an endomorphism of R) and he gave a commutative semiprimary local ring with a simple socle, which satisfies the principal extension property, but is not a PF ring (i.e. a selfinjective cogenerator ring). Though he did not succeed to determine whether it is QF-1 or not, the example suggests to us a question whether commutative perfect QF-1 rings are PF (equivalently QF by Osofsky [6, Theorem 3]) as similarly as in commutative noetherian strong QF-1 rings.The purpose of this paper is to give an affirmative answer to this question. We shall prove Theorem. Let R be a commutative perfect ring. Then every finitely generated faithful R-module is balanced if and only if R is a quasi-Frobenius ring.The proof of the Theorem uses the following lemma concerned with constructions of new nonbalanced modules.

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Epimorphic extensions of non-commutative rings

Cited by 22 publications

References 17 publications

The characteristic ring and the “best” way to adjoin a one

The characteristic ring and the “best” way to adjoin a one

Torsion theories over semihereditary rings

Commutative perfect 𝑄𝐹-1 rings

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