Polynomial rings over finite dimensional rings

Shock, Robert C.

doi:10.2140/pjm.1972.42.251

Cited by 51 publications

(20 citation statements)

References 3 publications

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“…It is well known that R is a semiprime ring if and only if R[x] is semiprime. It is an easy consequence of a theorem of Shock [11,Theorem 2.7] that the same holds for nonsingular rings. For PP-rings Armendariz [2] has shown that given a reduced ring R, R is a PP-ring if and only if R[x] is a PP-ring.…”

Section: Introductionmentioning

confidence: 91%

Rings with Projective Socle

Nicholson¹,

Watters²

1988

Proceedings of the American Mathematical Society

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ABSTRACT. The class of rings with projective left socle is shown to be closed under the formation of polynomial and power series extensions, direct products, and matrix rings. It is proved that a ring R has a projective left socle if and only if the right annihilator of every maximal left ideal is of the form ¡R, where / is an idempotent in R. This result is used to establish the closure properties above except for matrix rings. To prove this we characterise the rings of the title by the property of having a faithful module with projective socle, and show that if R has such a module, then so does M"(R).In fact we obtain more than Morita invariance.Also an example is given to show that eñe, for an idempotent e in a ring R with projective socle, need not have projective socle. The same example shows that the notion is not left-right symmetric. Introduction.We recall that a study of rings with projective socle and containing no infinite sets of orthogonal idempotents was make by Gordon [8]. In the same paper he showed that S = Soc rR is projective and essential if and only if S has zero right annihilator. More recently Baccella [4] has provided a number of necessary and sufficient conditions for R to have projective socle, conditions originally given by Manocha [9] when the socle was known to be essential. One such condition is that soc rR is nonsingular. However all these statements explicitly involve the socle. Our characterizations are somewhat different in that they involve either the maximal left ideals of R or modules with projective socles. Theorem 2.4 provides a number of conditions equivalent to the statement that R has a projective socle.As examples of rings with projective socles we have semiprime rings, nonsingular rings, and PP-rings. It is well known that R is a semiprime ring if and only if R[x] is semiprime. It is an easy consequence of a theorem of Shock [11, Theorem 2.7] that the same holds for nonsingular rings. For PP-rings Armendariz [2] has shown that given a reduced ring R, R is a PP-ring if and only if R[x] is a PP-ring. The same result holds for Baer rings and Burgess [6] remarks in his review of [2] that the Baer ring result is true for R\[x]} whilst for PP-rings it is false. Further, the polynomial results are not true if the restriction to reduced rings is removed. In Theorem 3.1 we show that if R has a projective socle, then so does R[x] (and i? [[a;]]), but the converse is false.In the final section we establish Morita invariance. Our proof of this uses an extension of an idea of Amitsur [1]. We call a module a PS-module if it has

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Section: Introductionmentioning

confidence: 91%

Rings with Projective Socle

Nicholson¹,

Watters²

1988

Proceedings of the American Mathematical Society

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“…Associated primes are well-known in commutative algebra for their important role in the primary decomposition, and has attracted a lot of attention in recent years. In [5], Brewer and Heinzer used localization theory to prove that, over a commutative ring R, the associated primes of the polynomial ring Motivated by the results in [2], [3], [7], [20], in this section, we first introduce the notion of weak associated primes, which is a generalization of associated primes. We next describe all weak associated primes of the generalized power series ring [[R S,≤ ]] in terms of the weak associated primes of the ring R. …”

Section: Weak Associated Primesmentioning

confidence: 99%

Special Weak Properties of Generalized Power Series Rings

Ouyang¹

2012

Journal of the Korean Mathematical Society

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Abstract. Let R be a ring and nil(R) the set of all nilpotent elements of R. For a subset X of a ring R, we define N R (X) = {a ∈ R | xa ∈ nil(R) for all x ∈ X}, which is called a weak annihilator of X in R. A ring R is called weak zip provided that for any subset, and a ring R is called weak symmetric if abc ∈ nil(R) ⇒ acb ∈ nil(R) for all a, b, c ∈ R. It is shown that a generalized power series ring [[R S,≤ ]] is weak zip (resp. weak symmetric) if and only if R is weak zip (resp. weak symmetric) under some additional conditions. Also we describe all weak associated primes of the generalized power series ring [[R S,≤ ]] in terms of all weak associated primes of R in a very straightforward way.

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“…Thus y-g (F P kD)kG, and (13) is proved. Put M = kD/Y n kD, so V = M 9kDkG by (13), and A/ is irreducible. By Lemma 7.2, 0 # / = AnnkZ(M).…”

Section: Thus H¡/h¡_ X Is Infinite Cyclic and Pf = Bh¡_vmentioning

confidence: 99%

Primitive group rings and Noetherian rings of quotients

Brookes¹,

Brown²

1985

Trans. Amer. Math. Soc.

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Abstract.Let k be a field, and let G be a countable nilpotent group with centre Z. We show that the group algebra kG is primitive if and only if k is countable, G is torsion free, and there exists an abelian subgroup A of G, of infinite rank, with A C\ Z = 1. Suppose now that G is torsion free. Then kG has a partial quotient ring Q = kG(kZ)'1. The above characterisation of the primitivity of kG is intimately connected with the question: When is Q a Noetherian ring? We determine this for those groups G, as above, all of whose finite rank subgroups are finitely generated. In this case, Q is Noetherian if and only if G has no abelian subgroup A of infinite rank with A n Z = 1.

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Polynomial rings over finite dimensional rings

Cited by 51 publications

References 3 publications

Rings with Projective Socle

Rings with Projective Socle

Special Weak Properties of Generalized Power Series Rings

Primitive group rings and Noetherian rings of quotients

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