منابع مشابه
Of Integral Domains
The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by ideal t-multiplication. This paper investigates ring-theoretic properties of an integral domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup. Contexts (including Lipman and Sally-Vasconcelos stability) that suit best t-mult...
متن کاملClass Semigroups of Integral Domains
This paper seeks ring-theoretic conditions of an integral domain R that reflect in the Clifford property or Boolean property of its class semigroup S(R), that is, the semigroup of the isomorphy classes of the nonzero (integral) ideals of R with the operation induced by multiplication. Precisely, in Section 3, we characterize integrally closed domains with Boolean class semigoup; in this case, S...
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My son who is in the 4 grade is learning about prime numbers and cancelling prime numbers in order to reduce fractions into lowest forms. I have told him that every number (positive integer) can be expressed as a product of primes, and surely along the road, his teachers will confirm this. We will consider this property in integral domains. We say that a divides b in the domain R, and write a |...
متن کاملFactorization in Integral Domains II
Theorem 1.1 (Rational roots test). Let f = anx n + · · · + a0 ∈ Z[x] be a polynomial of degree n ≥ 1 with integer coefficients and nonzero constant term a0, and let p/q ∈ Q be a rational root of f such that the fraction p/q is in lowest terms, i.e. gcd(p, q) = 1. Then p divides the constant term a0 and q divides the leading coefficient an. In particular, if f is monic, then a rational root of f...
متن کاملUnique Factorization in Integral Domains
Throughout R is an integral domain unless otherwise specified. Let A and B be sets. We use the notation A ⊆ B to indicate that A is a subset of B and we use the notation A ⊂ B to mean that A is a proper subset of B. The group of elements in R which have a multiplicative inverse (the group of units of R) is denoted R×. Since R has no zero divisors cancellation holds. If a, b, c ∈ R and a 6= 0 th...
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ژورنال
عنوان ژورنال: Journal of Algebra
سال: 1984
ISSN: 0021-8693
DOI: 10.1016/0021-8693(84)90044-9