Semirigid GCD domains II

Zafrullah, Muhammad

doi:10.1142/s0219498822501614

Cited by 2 publications

(2 citation statements)

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“…(1) Let D be a semirigid PSP domain then G(D) + or m(D) the monoid of nonzero principal ideals of D is a v-factorial monoid. Of course if D is a GCD domain and semirigid we get a semirigid GCD domain of[43] where m(D) is a factorial monoid. Example 3.7 of[43] serves as an example of a semirigid Schreier domain.…”

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confidence: 99%

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Riesz and pre-Riesz monoids

Zafrullah¹

2021

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Call a directed partially ordered cancellative divisibility monoid M a Riesz monoid if for all x, y 1 , y 2 ≥ 0 in M, x ≤ y 1 + y 2 ⇒ x = x 1 + x 2 where 0 ≤ x i ≤ y i . We explore the necessary and sufficient conditions under which a Riesz monoid M with M + = {x ≥ 0|x ∈ M } = M generates a Riesz group and indicate some applications. We call a directed p.o. monoid M Πpre-Riesz if M + = M and for all x 1 , x 2 , ..., xn ∈ M , glb(x 1 , x 2 , ..., xn) = 0 or there is r ∈ Π such that 0 < r ≤ x 1 , x 2 , ..., xn, for some subset Π of M.We explore examples of Π-pre-Riesz monoids of * -ideals of different types. We show for instance that if M is the monoid of nonzero (integral) ideals of a Noetherian domain D and Π the set of invertible ideals, M is Π-pre-Riesz if and only D is a Dedekind domain. We also study factorization in pre-Riesz monoids of a certain type and link it with factorization theory of ideals in an integral domain.

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confidence: 99%

“…Of course if D is a GCD domain and semirigid we get a semirigid GCD domain of[43] where m(D) is a factorial monoid. Example 3.7 of[43] serves as an example of a semirigid Schreier domain. Of course a UFD is a PSP domain and can be treated as a semirigid PSP domain.…”

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confidence: 99%