منابع مشابه
Kaplansky-type Theorems in Graded Integral Domains
It is well known that an integral domain D is a UFD if and only if every nonzero prime ideal of D contains a nonzero principal prime. This is the so-called Kaplansky’s theorem. In this paper, we give this type of characterizations of a graded PvMD (resp., G-GCD domain, GCD domain, Bézout domain, valuation domain, Krull domain, π-domain).
متن کاملTwo theorems of Glaisher and Kaplansky
We give a new proof of a recent theorem of Kaplansky and use it to revive an old, seemingly forgotten result of Glaisher. It is well known that, for any n > 0, the prime numbers of the form x +ny can be described by congruence conditions if and only if n is one of Euler’s convenient numbers [4, p. 62]. Since 1, 2, 4, 8 and 16 are the only convenient powers of 2, the following theorem of Kaplans...
متن کاملNoncommutative Generalizations of Theorems of Cohen and Kaplansky
This paper investigates situations where a property of a ring can be tested on a set of “prime right ideals.” Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff every “prime right ideal” is finitely generated (resp. principal), where the phrase “prime right ideal” can be interpreted in one of many different ways. We...
متن کاملOn a Question of Kaplansky Ii
There is a question attributed to Irving Kaplansky concerning the solvability of the quadratic equation x2 − py2 = a in the case that the prime p = a2 + (2b)2. This question was answered in the affirmative by Mollin [1], although according to [3], this result is implicit in the work of Gauss and Legendre. The proof appearing in [1] was later simplified in [4], and it was also shown therein that...
متن کاملOn a Conjecture of Kaplansky
PROOF. Let A be a C*-algebra, ' a derivation of A. It is enough to show that the derivation is continuous on the self-adjoint portion As of A. Therefore if it is not continuous, by the closed graph theorem there is a sequence \xn\ (xn 4= 0) in As such that xn -»0 and xn -> a + ίέ(φθ), where a and b are self-adjoint. First, suppose that a =f= 0 and there exists a positive number λ(> 0) in the sp...
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ژورنال
عنوان ژورنال: Kyungpook mathematical journal
سال: 2011
ISSN: 1225-6951
DOI: 10.5666/kmj.2011.51.3.339