Recent results on weakly factorial domains

In this paper, we will survey recent results on weakly factorial domains base on the results of [11, 13, 14]. Let D be an integral domain, X be an indeterminate over D, d ∈ D, R = D[X, d X ] be a subring of the Laurent polynomial ring D[X, 1 X ], Γ be a nonzero torsionless commutative cancellative monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. Among other things, we show that R is a weakly factorial domain if and only if D is a weakly factorial GCD-domain and d = 0, d is a unit of D or d is a prime element of D. We also show that if char(D) = 0 (resp., char(D) = p > 0), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0, 0, 0, . . . ) (resp., (0, 0, 0, . . . ) except p).


Introduction
Let D be an integral domain with quotient field K, X be an indeterminate over D, Γ be a nonzero torsionless commutative cancellative monoid (written additively) with quotient group G, and D[Γ] be the semigroup ring of Γ over D. A nonzero nonunit element p ∈ D is called a primary element if p|ab for each a, b ∈ D implies that p|a or p|b n for some positive integer n; equivalently, pD is a primary ideal.We say that D is a unique factorization domain (UFD) (resp., weakly factorial domain (WFD)) if each nonzero nonunit of D can be written as a finite product of prime (resp.primary) elements.Clearly, a prime element is primary, and hence a UFD is a WFD.The concept of a WFD was first introduced by Anderson and Mahaney [3].Chang introduced the notion of weakly factorial semigroups [11] in order to study when D[Γ] is a WFD.A nonzero nonunit element s ∈ Γ is primary if, for each a, b ∈ Γ, a + b ∈ s + Γ implies that a ∈ s + Γ or nb ∈ s + Γ for some positive integer n.It is clear that s ∈ Γ is primary if and only if s + Γ is a primary ideal.We say that Γ is a weakly factorial semigroup if each nonunit of Γ is a finite sum of primary elements.
Chang has worked on WFDs [6-8, 11, 13, 14] and its generalization to rings with zero divisors [10,12].In this paper, among them, we survey the recent results of [11,13,14].In Section 1, for easy reference, we review some definitions and preliminary results on toperations, monoinds, semigroup rings, and weakly Krull domains.Let In Section 2, we show that R is a weakly factorial domain if and only if D is a weakly factorial GCD-domain and d = 0, d is a unit of D or d is a prime element of D. We also show that if D is a weakly factorial GCD-domain, p is a prime element of D, and n ≥ 2 is an integer, X ] is an almost weakly factorial domain with Cl(D[X, p n X ]) = Z n , the cyclic group of order n.In Section 3, we completely characterize when D[Γ] is a WFD.We first study the notion of weakly factorial GCD-semigroups.Then, among other things, we prove that if char(D) = 0, then D[Γ] is a WFD if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0, 0, 0, . . .).We also show that if char(D) = p > 0, then D[Γ] is a WFD if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0, 0, 0, . . . ) except p.However, we omit the proofs of the results; so the reader who is interested in the proofs can refer to [11,13,14].

Definitions and preliminary results
In this section, we review some definitions and preliminary results on the t-operations, semigroups, semigroup rings, and weakly Krull domains.

The t-operations
Let D be an integral doman with quotient field K and F(D) be the set of nonzero fractional ideals of D.
Hence, we can define the vand t-operations as follows: I v = (I −1 ) −1 , and is a prime ideal; each prime ideal minimal over a t-ideal is a t-ideal (which implies that every heightone prime ideal is a t-ideal); and D = P∈t-Max(D) D P .We say that D is of finite t-character if every nonzero nonuint of D is contained in only finitely many maximal t-ideals of D.
An I ∈ F(D) is said to be t-invertible if (II −1 ) t = D. Let T (D) be the group of t-invertible fractional t-ideals of D under I * J = (IJ) t , and Prin(D) be its subgroup of principal fractional ideals.Then Cl(D) = T (D)/Prin(D), called the t-class group of D, is an abelian group.We say that D is a Prüfer v-multiplication domain (PvMD) if each nonzero finitely generated ideal of

Semigroups
Let Γ be a nonzero torsionless grading monoid, i.e., a nonzero torsionless commutative cancellative monoid (written additively), and Γ = {a − b | a, b ∈ Γ} be the quotient group of Γ; so Γ is a torsionfree abelian group and Γ is given a total order compatible with the monoid operation [22, page 123].Let G be a torsionfree abelian group.We say that G is of type (0, 0, 0, . ..) if G satisfies the ascending chain condition on its cyclic subgroups (equivalently, for each nonzero element g ∈ G, there exists a largest positive integer n g such that n g x = g is solvable in G) [17,Theorem 14.10].For a prime number p, G is said to be of type (0, 0, 0, . ..) except p if G satisfies the following two conditions; for each nonzero element g ∈ G, (i) the number of prime numbers dividing g is finite and (ii) for each prime number q p, q n does not divide g for some positive integer n.The notion of type (0, 0, 0, . ..) except p was introduced by Matsuda [23,25] in order to study when K[G], where K is a field with char(K) = p, is a generalized Krull domain.Clearly, a torsionfree abelian group of type (0, 0, 0, . ..) is of type (0, 0, 0, . ..) except p for all prime numbers p.As in the case of integral domains, we can define the t-operation on Γ.For more on definitions and basic results (e.g., maximal t-ideals, class groups, weakly factorial semigroups), see [20,Chapter 11], [15], or [11].[17,Corollary 3.4].Each nonzero element of the form aX s ∈ D[Γ] is said to be homogeneous with deg(aX s ) = s.Let H be the set of nonzero homogeneous elements of D , and hence D[Γ] H is a completely integrally closed GCD-domain [18,Theorem 6.4].For more on semigroup rings, the reader can refer to [17].

Weakly Krull domains
Let X 1 (D) be the set of height-one prime ideals of an integral domain D. We say that D is a weakly Krull domain if D = P∈X 1 (D) D P and this intersection has finite character, i.e., each nonzero nonunit of D is a unit in D P except finitely many primes in X 1 (D).

Weakly factorial generalized Rees rings
Let D be an integral domain, I be a proper ideal of D, and t be an indeterminate over D. Then R = D[tI, t −1 ] is a subring of D[t, t −1 ], called the generalized Rees ring of D with respect to I. In [27], Whithman proved that if I is finitely generated, then R is a UFD if and only if D is a UFD and t −1 is a prime element of R. Let I = dD for d ∈ D and t −1 = X; so R = D[X, d X ].In [1], the authors studied several kinds of divisibility properties of R including Krull domains, UFDs, and GCD-domains.In this section we study when Then the following statements are equivalent.

d
X is irreducible (resp., prime, primary) in R.

D is a weakly factorial GCD-domain and d = 0, d is a unit of D, or d is a prime element of D.
A  Theorems 8 and 16].The next result is an AWFD analog.
Corollary 2.7.[13, Corollary 9] Let D be a weakly factorial GCD-domain, p be a prime element of D, and n ≥ 2 be an integer.Then R = D[X, p n X ] is an AWFD with Cl(R) = Z n .Let V be a rank-one nondiscrete valuation domain, y be an indeterminate over V, and Then D is a weakly factorial GCD-domain and y is a prime of D. Thus, by Theorem 2.4 and Corollary 2.

Weakly factorial semigroup rings
Let D be an integral domain with quotient field K, Γ be a torsionless grading monoid with quotient group G, D[Γ] be the semigroup ring of Γ over D, and H be the set of nonzero homgeneous elements of D [Γ].
In order to study when D[Γ] is a UFD, we first need the notion of factorial semigroups.As in the domain case, an α ∈ Γ is called a prime element if α + Γ is a prime ideal of Γ, and we say that Γ is a factorial semigroup if each nonunit of Γ can be written as a finite sum of prime elements of Γ. Theorem 3.1.D[Γ] is a UFD if and only if D is a UFD, Γ is a factorial semigroup, and the quotient group of Γ is of type (0, 0, 0, . . .).
Proof.[18, Theorem 7.17 and Lemma 7.15].Theorem 3.1 is the motivation of the results in this section, i.e., we completely characterize when D[Γ] is a WFD.

α is primary in Γ.
It is well known that D is a GCD-domain if and only if aD ∩ bD is principal, if and only if (a, b) v is principal for each pair 0 a, b ∈ D. Similarly, note that if Γ is a GCD-semigrup, then gcd(α, β) It is clear that a factorial semigroup is a GCD-semigroup.Also, prime elements are primary, and hence a factorial semigroup is a weakly factorial GCD-semigroup.However, the next example shows that a weakly factorial GCD-semigroup whose quotient group is of type (0, 0, 0, . . . ) need not be a factorial semigroup.1. Γ is a torsionless grading monoid with quotient group G.

. ). Then D[Γ] is a WFD if and only if D is a weakly factorial GCD-domain and Γ is a weakly factorial GCD-semigroup.
Since a factorial semigroup is a weakly factorial GCD semigroup, by Theorem 3.5, we have Corollary 3.6.[11, Corollary 10] Assume that G is of type (0, 0, 0, . . . ) and Γ is a factorial semigroup.Then D[Γ] is a WFD if and only if D is a weakly factorial GCD-domain.
Set H α = N 0 for each α, and let Γ = α H α .Then K[Γ] = K[{X α }] is a factorial domain, and hence Γ is a factorial semigroup whose quotient group is of type (0, 0, 0, . . .).Thus, by Corollary 3.6, we have The next example shows that Theorem 3.5 is not true when G is not of type (0, 0, 0, . . .).Example 3.9.[11, Example 13] Let R be the field of real numbers and Γ be the additive semigroup of nonnegative rational numbers.

ITM
Web of Conferences 20, 01001 (2018) https://doi.org/10.1051/itmconf/20182001001ICM 2018 1.3 Semigroup rings For a nonzero torsionless commutative cancellative monoid Γ and an integral domain D, let D[Γ] be the semigroup ring of Γ over D. Then D[Γ] is an integral domain [17, Theorem 8.1] and each nonzero element f ∈ D[Γ] can be written uniquely as and we say that D is a UMT-domain if each upper to zero in D[X] is a maximal t-ideal of D[X].It is known that D is a PvMD if and only if D is an integrally closed UMT-domain [21, Proposition 3.2].Proposition 1.2.The following statements are equivalent for D. 1. D[X] is a weakly Krull domain.2. D[X, X −1 ] is a weakly Krull domain.3. D is a weakly Krull UMT-domain.Proof.[2, Propositions 4.7 and 4.11].Clearly, Krull domains are weakly Krull domains.Recall that D is a UFD if and only if D is a Krull domain and Cl(D) = {0} [16, Proposition 6.1].Theorem 1.3.D is a WFD if and only if D is a weakly Krull domain and Cl(D) = {0}.Proof.[5, Theorem].An almost weakly factorial domain (AWFD) is an integral domain D in which for each 0 d ∈ D, there is an integer n = n(d) ≥ 1 such that d n can be written as a finite product of primary elements of D. It is known that D is an AWFD if and only if D is a weakly Krull domain with Cl(D) torsion [4, Theorem 3.4]; hence UFD ⇒ WFD ⇒ AWFD ⇒ weakly Krull domain.It is easy to see that if N is a multiplicative subset of a weakly Krull domain (resp., WFD, an AWFD), then D N satisfies the corresponding property.ITM Web of Conferences 20, 01001 (2018) https://doi.org/10.1051/itmconf/20182001001ICM 2018

Corollary 3 . 7 .
[11,  Corollary 11]  Let {X α } be a nonempty set of indeterminates over D. Then D[{X α }] is a WFD if and only if D is a weakly factorial GCD-domain.Example 3.8.Let Z be the ring of integers and Γ = {m + n √ 2 | m, n ∈ Z and m + n √ 2 ≥ 0}.Then Z[Γ] is a WFD but not a UFD by Theorems 3.1, 3.5 and Example 3.4.
3. d is a nonunit (resp., prime, primary) in D. Proof.For the properties of irreducible and prime, see [1, Proposition 1].For the property of primary, see [13, Proposition 2].The next lemma can be easily proved by Proposition 1.2 which is necessary for the proof of Thorem 2.4.
Let 0 a ∈ D. It is easy to see that a is primary if and only if √ aD is a maximal t-ideal [7, Lemma 2.1].Hence, by Proposition 2.1, we have Corollary 2.2.[13,Corollary 3] Let d ∈ D be a nonzero nonunit and R= D[X, d X ].If dD is primary in D, then √ XR is a maximal t-ideal of R and (X, d X ) v = R.
ring of Krull type is a PvMD of finite t-character [19, Theorem 7].It is known that D is a ring of Krull type if and only if D[X] is a ring of Krull type, or equivalently, D[X, 1 X ] is a ring of Krull type (cf.[19, Propositions 9 and 12]).generalized Krull domain is an integral domain D such that (i) D P is a valuation domain for all P ∈ X 1 (D), (ii) D = P∈X 1 (D) D P , and (iii) this intersection has finite character.Clearly, D is a generalized Krull domain if and only if D is a weakly Krull PvMD, if and only if D is a ring of Krull type with t-dim(D) = 1.Hence, by Lemma 2.3 andTheorem 2.5, we have ITM Web of Conferences 20, 01001 (2018) https://doi.org/10.1051/itmconf/20182001001ICM 2018 Corollary 2.6.[13, Corollary 8] Let d ∈ D and R = D[X, d X ].Then R is a generalized Krull domain if and only if D is a generalized Krull domain.
[13,rem 2.5.[13,Theorem7]Letd∈ D and R = D[X, d X ].Then Ris a ring of Krull type if and only if D is a ring of Krull type.A Thus, Γ is a GCD-semigroup if and only if k i=1 (α i + Γ) is principal, if and only if ( k i=1 (α i + Γ)) v is principal for any α 1 , . . ., α k ∈ Γ (cf.[20, Exercise 1 (page 117) and Theorem 11.5]).The next result is the semigroup analog of [3, Theorem 18] that a WFD D is a GCD-domain if and only if D P is a valuation domain for each height-one prime ideal P of D. Proposition 3.3.[11, Proposition 5] Suppose that Γ is a weakly factorial semigroup.Then Γ is a GCD-semigroup if and only if α + Γ ⊆ β + Γ or β + Γ ⊆ α + Γ for all primary elements α and β of Γ with √