By Scott T. Chapman

ISBN-10: 0824723279

ISBN-13: 9780824723279

ISBN-10: 1420028243

ISBN-13: 9781420028249

------------------Description-------------------- The research of nonunique factorizations of parts into irreducible parts in commutative earrings and monoids has emerged as an self sufficient zone of analysis merely during the last 30 years and has loved a re

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**Sample text**

Mn . In this case, R0 is a subring of R, and (R0 )∗ is a saturated multiplicatively closed subset of R. Thus U (R) = U (R0 ). Also, the set S = { 0 = x ∈ R | x is homogeneous } = ∪n∈Z+ (Rn )∗ of nonzero homogeneous elements of R is a saturated multiplicatively closed subset of R. Then RS is a Z-graded integral domain via deg(a/s) = deg(a) − deg(s) for all a, s ∈ S. We will call RS the homogeneous quotient ﬁeld of R; each nonzero homogeneous element of R is a unit. Note that (RS )0 is a ﬁeld, and moreover, RS = (RS )0 [t, t−1 ] for any homogeneous t ∈ RS of smallest positive degree (such a t is transcendental over (RS )0 ), and hence RS is a PID.

Let A ⊆ B be an extension of integral domains and R = A+XB[X]. (a) If B is a ﬁeld, then R is an HFD if and only if A is a ﬁeld. (b) If A is a ﬁeld, then R is an HFD if and only if B is integrally closed. (c) If B is a UFD, U (B) ∩ A = U (A), and each irreducible element of A is also irreducible in B, then R is an HFD. Proof. 1]. 4]. For an atomic integral domain R, the elasticity of R is ρ(R) = sup{ m/n | x1 · · · xm = y1 · · · yn for irreducible xi , yj ∈ R }. ) Then 1 ≤ ρ(R) ≤ ∞, and ρ(R) = 1 if and only if R is an HFD.

So a is ∗-pseudo-prime. (2)=⇒(3) Clear. (3)=⇒(2) Let a be ∗-pseudo-irreducible. Then a has a ∗-comaximal factorization a = a1 · · · an where each ai is ∗-pseudo-prime. Since a is ∗-pseudo-irreducible, n = 1, that is, a = a1 is ∗-pseudo-prime. (2)=⇒(4) Suppose that x1 · · · xn = y1 · · · ym are two complete ∗-comaximal factorizations. By hypothesis, each xi , yj is ∗-pseudo-prime. Now x1 ∗-pseudo-prime and x1 |y1 · · · ym implies x1 |yi for some i which we can take to be 1. Likewise yi |xj for some j.

### Arithmetical Properties of Commutative Rings and Monoids by Scott T. Chapman

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