Cancellation ideals of a ring extension
We study properties of cancellation ideals of ring extensions. Let R ⊆ S be a ring extension. A nonzero S-regular ideal I of R is called a (quasi)-cancellation ideal of the ring extension R ⊆ S if whenever IB = IC for two S-regular (finitely generated) R-submodules B and C of S, then B = C. We show...
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nasplib_isofts_kiev_ua-123456789-1887222025-02-09T15:55:12Z Cancellation ideals of a ring extension Tchamna, S. We study properties of cancellation ideals of ring extensions. Let R ⊆ S be a ring extension. A nonzero S-regular ideal I of R is called a (quasi)-cancellation ideal of the ring extension R ⊆ S if whenever IB = IC for two S-regular (finitely generated) R-submodules B and C of S, then B = C. We show that a finitely generated ideal I is a cancellation ideal of the ring extension R ⊆ S if and only if I is S-invertible. 2021 Article Cancellation ideals of a ring extension / S. Tchamna // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 1. — С. 138–146. — Бібліогр.: 7 назв. — англ. 1726-3255 DOI:10.12958/adm1424 2020 MSC: 13A15, 13A18, 13B02 https://nasplib.isofts.kiev.ua/handle/123456789/188722 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
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We study properties of cancellation ideals of ring extensions. Let R ⊆ S be a ring extension. A nonzero S-regular ideal I of R is called a (quasi)-cancellation ideal of the ring extension R ⊆ S if whenever IB = IC for two S-regular (finitely generated) R-submodules B and C of S, then B = C. We show that a finitely generated ideal I is a cancellation ideal of the ring extension R ⊆ S if and only if I is S-invertible. |
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Tchamna, S. Cancellation ideals of a ring extension Algebra and Discrete Mathematics |
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Cancellation ideals of a ring extension |
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Cancellation ideals of a ring extension |
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Cancellation ideals of a ring extension |
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Cancellation ideals of a ring extension |
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Cancellation ideals of a ring extension |
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cancellation ideals of a ring extension |
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Cancellation ideals of a ring extension / S. Tchamna // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 1. — С. 138–146. — Бібліогр.: 7 назв. — англ. |
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“adm-n3” — 2021/11/8 — 20:27 — page 138 — #140
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 32 (2021). Number 1, pp. 138–146
DOI:10.12958/adm1424
Cancellation ideals of a ring extension
S. Tchamna
Communicated by E. I. Zelmanov
Abstract. We study properties of cancellation ideals of ring
extensions. Let R ⊆ S be a ring extension. A nonzero S-regular ideal
I of R is called a (quasi)-cancellation ideal of the ring extension
R ⊆ S if whenever IB = IC for two S-regular (finitely generated)
R-submodules B and C of S, then B = C. We show that a finitely
generated ideal I is a cancellation ideal of the ring extension R ⊆ S
if and only if I is S-invertible.
1. Introduction and background
Throughout this article, we assume that all rings are commutative with
identity. The notion of cancellation ideal for a ring has been studied in [1]
and [2]. An ideal I of a ring R is called cancellation ideal if whenever
IB = IC for two ideals B and C of R, then B = C [2]. A finitely generated
ideal is a cancellation ideal if and only if for each maximal ideal M of R,
IM is a regular principal ideal of RM [1, Theorem 1]. D.D Anderson and
D.F Anderson used the notion of cancellation ideal to characterize Prüfer
domain. A ring R is a Prüfer domain if and only if every finitely generated
nonzero ideal of R is a cancellation ideal[1, Theorem 6]. In this paper,
we study the notion of cancellation ideal for ring extensions; which is a
generalization of the notion of cancellation ideal for rings. Let R ⊆ S be a
ring extension, and let A be an R-submodule of S. The R-submodule A is
said to be S-regular ifAS = S[5, Definition 1, p. 84]. For two R-submodules
E, F of S, denote by [E : F ] the set of all x ∈ S such that xF ⊆ E.
2020 MSC: 13A15, 13A18, 13B02.
Key words and phrases: ring extension, cancellation ideal, pullback diagram.
https://doi.org/10.12958/adm1424
“adm-n3” — 2021/11/8 — 20:27 — page 139 — #141
S. Tchamna 139
An R-submodule A of S is said to be S-invertible, if there exists an R-
submodule B of S such that AB = R[5, Definition 3, p 90]. In this case,
we write B = A−1, and A−1 = [R : A] = {x ∈ S : xA ⊆ R} [5, Remark
1.10, p. 90]. For the R-submodule A of S, and for a multiplicative subset
τ of R, we denote by A[τ ] the set of all x ∈ S such that tx ∈ A for some
t ∈ τ . If p is a prime ideal of R, and τ = R \ p, then A[p] denotes the
set of all x ∈ S such that tx ∈ A for some t ∈ τ . The set A[τ ] is called
the saturation of A by τ . Properties of the saturation of a submodule are
studied in [5, p. 18] and [6].
An S-regular ideal I of R is called (quasi)-cancellation ideal of the
ring extension R ⊆ S if whenever IB = IC for two S-regular (finitely
generated) R-submodules B and C of S, then B = C. In section 2,
we study properties of (quasi)-cancellation ideals of ring extensions. In
Proposition 2.4, we prove that a finitely generated S-regular ideal I of
R is a cancellation ideal if and only it is a quasi-cancellation ideal. In
Theorem 2.12, we show that for an S-regular finitely generated ideal I of
R, the followings are equivalent:
(1) I is a cancellation ideal of the ring extension R ⊆ S.
(2) I is an S-invertible ideal of R.
(3) IR[X] is a cancellation ideal of the ring extension R[X] ⊆ S[X].
Remark 1.1. Let R ⊆ S be a ring extension, and let A,B be two R-
submodules of S. Then A = B if and only if A[m] = B[m] for each maximal
ideal m of R. In fact, if A = B, then it clear that A[m] = B[m] for each
maximal ideal m of R. Conversely, if A[m] = B[m] for each m ∈ M, where
M is the set of all maximal ideals of R, then by [5, Remark 5.5, p. 50],
we have A = ∩m∈MA[m] = ∩m∈MBm = B.
Let R ⊆ S and L ⊆ T be two ring extensions, and consider the following
commutative diagram
R L
S T
Ψ
α
where kerΨ is an ideal of R, Ψ : S −→ T is surjective, the restriction
α : R −→ L of Ψ is also surjective and the vertical mappings are inclusions.
When kerΨ is a maximal ideal of S, the previous commutative diagram
is called a pullback diagram a type �. Pullback diagrams of type � are
studied by S. Gabelli and E. Houston in [4].
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140 Cancellation ideals of a ring extension
Lemma 1.2. Consider the above pullback diagram of type �. If A,B are
two S-regular ideals of R such that Ψ(A) = Ψ(B), then A = B.
Proof. Let A,B be two S-regular ideals of R such that Ψ(A) = Ψ(B).
By [7, Remark 1.1], we have kerΨ ⊆ A and kerΨ ⊆ B. Let a ∈ A. Then
there exists b ∈ B such that Ψ(a) = Ψ(b). Hence a− b ∈ kerΨ ⊆ B. Thus
a ∈ B. This shows that A ⊆ B. With the same argument, B ⊆ A. Thus
A = B.
2. Cancellation ideals of ring extensions
In this section, we define and study properties of cancellation ideals of
ring extensions.
Definition 2.1. Let R ⊆ S be a ring extension. A nonzero S-regular ideal
I of R is called a (quasi)-cancellation ideal of the ring extension R ⊆ S if
whenever IB = IC for two S-regular (finitely generated) R-submodules B
and C of S, then B = C.
The following proposition studies cancellation ideals in pullback diagram
of type �. In this article, the Jacobson radical of a ring is denoted Jac(R).
Proposition 2.2. Suppose that the following diagram
R L
S T
Ψ
is a pullback diagram of type � such that kerΨ ⊆ Jac(R). Then an S-
regular ideal I of R is a cancellation ideal of the extension R ⊆ S if and
only if Ψ(I) is a cancellation ideal of the extension L ⊆ T .
Proof. Suppose that I is a cancellation ideal of the extension R ⊆ S. Since
IS = S, we have Ψ(I)Ψ(S) = Ψ(S). It follows that Ψ(I)T = T . Hence
Ψ(I) is a T -regular ideal of L. Let E and F be two T -regular L-submodules
of T such that Ψ(I)E = Ψ(I)F . Let B = Ψ−1(E) and C = Ψ−1(F ). Then
by [7, Lemma 2.8(1)] B and C are two S-regular ideals of R. Furthermore,
E = Ψ(B) and F = Ψ(C) since Ψ is surjective. It follows from the equality
Ψ(I)E = Ψ(I)F that Ψ(I)Ψ(B) = Ψ(I)Ψ(C). Hence Ψ(IB) = Ψ(IC).
Furthermore, (IB)S = IS = S and (IC)S = IS = S. Therefore, by
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S. Tchamna 141
Lemma 1.2, we have IB = IC. Hence B = C since I is a cancellation
ideal of the extension R ⊆ S. It follows that E = Ψ(B) = Ψ(C) = F .
This shows that Ψ(I) is a cancellation ideal of the extension L ⊆ T .
Conversely, suppose that Ψ(I) is a cancellation ideal of the extension
L ⊆ T . Let B and C be two S-regular R-submodules of S such that IB =
IC. Then Ψ(I)Ψ(B) = Ψ(I)Ψ(C). Since BS = S, we have Ψ(B)T = T .
Hence Ψ(B) is a T -regular ideal of L. With the same argument, Ψ(C)
is a T -regular ideal of L. It follows that Ψ(B) = Ψ(C) since Ψ(I) is a
cancellation ideal of the extension L ⊆ T . Therefore, by Lemma 1.2, we
have B = C. This shows that I is a cancellation ideal of the extension
R ⊆ S.
In the next proposition, we give a characterization of a cancellation ideal
of a ring extension. This result is an analogue of [3, Proposition 2.1, p.
10] in the case of cancellation ideal of a ring.
Proposition 2.3. Let R ⊆ S be a ring extension, and let I be an S-regular
ideal of R. The following statements are equivalent.
(1) I is a (quasi)-cancellation ideal of the ring extension R ⊆ S.
(2) [IJ : I] = J for any S-regular (finitely generated) R-submodule J of
S.
(3) If IJ ⊆ IK for two S-regular (finitely generated) R-submodules J
and K of S, then J ⊆ K.
Proof. (1) ⇒ (2) Suppose that I is a cancellation ideal of the extension
R ⊆ S, and let J be an S-regular R-submodule of S. The containment
J ⊆ [IJ : I] is always true. Let x ∈ [IJ : I]. Then xI ⊆ IJ . It follows that
(x, J)I ⊆ IJ , where (x, J) is the R-submodule of S generated by x and J .
Therefore, (x, J)I = IJ since the containment IJ ⊆ (x, J)I is always true.
Furthermore, (x, J) is an S-regular R-submodule of S since J ⊆ (x, J). It
follows from the definition of a cancellation ideal that (x, J) = J . This
shows that x ∈ J , and thus [IJ : I] ⊆ J . Therefore [IJ : I] = J .
(2) ⇒ (3) Suppose that the statement (2) is true. Let J and K be
two S-regular R-submodules of S. Then by (2), we have [IK : I] = K. If
IJ ⊆ IK, then J ⊆ [IK : I] = K.
(3) ⇒ (1) This implication is obvious.
Proposition 2.4. Let R ⊆ S be a ring extension, and let I be a finitely
generated S-regular ideal of R. Then I is a cancellation ideal of R ⊆ S if
and only if I is a quasi-cancellation ideal of R ⊆ S.
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142 Cancellation ideals of a ring extension
Proof. Let I be a finitely generated S-regular ideal of R. If I is a can-
cellation ideal of the extension R ⊆ S, then obviously I is an quasi-
cancellation ideal of the extension R ⊆ S. Conversely, suppose that I is
a quasi-cancellation ideal of the extension R ⊆ S. Let a1, . . . , an ∈ R be
a set of generators of I. Let B,C be two S-regular R-submodules of S
such that IB ⊆ IC. Let b ∈ B. Then bI ⊆ IC. So, for 1 6 i 6 n, we have
bai =
∑k
j=1 ajcij with cij ∈ C for 1 6 j 6 k. Furthermore, since CS = S,
there exist u1, . . . , uℓ ∈ C and s1, . . . , sℓ ∈ S such that u1s1+· · ·+uℓsℓ = 1.
Let C ′ be the R-submodule of S generated by the elements of the set
{u1, . . . , un, cij : 1 6 i 6 n, 1 6 j 6 k}. Let B0 be the R-submodule of S
generated by b. Then (B0 + (u1, . . . , un)R) I ⊆ IC ′. It follows from the
the equivalence (1) ⇔ (3) of Proposition 2.3 that B0+(u1, . . . , un)R ⊆ C ′
since B0 + (u1, . . . , un) and C ′ are finitely generated S-regular ideal of S.
Therefore, b ∈ C ′ ⊆ C. Hence B ⊆ C since b was arbitrary chosen in B.
This shows that I is a cancellation ideal of the extension R ⊆ S.
Lemma 2.5. Let R ⊆ S be a ring extension, and let u1, . . . , uℓ ∈ S.
Define the sets E = (u1, . . . , uℓ)R[p] and A = (u1, . . . , uℓ)R, where p is a
prime ideal of R. For any ideal I of R, we have:
(1) (AI)[p] = (EI)[p]. In particular, A[p] = E[p].
(2) (EI)[p] = (EI[p])[p].
Proof. (1) First, observe that AI ⊆ EI. So (AI)[p] ⊆ (EI)[p]. Let x ∈
(EI)[p]. Then there exists t ∈ R \ p such that tx ∈ EI. Therefore, tx =
∑n
i=1 eixi for some ei ∈ E and xi ∈ I, 1 6 i 6 n. For each 1 6 i 6 n,
write ei =
∑ℓ
j=1 ujyij with yij ∈ R[p] for 1 6 j 6 ℓ. Let sij ∈ R \ p such
that sijyij ∈ R, si =
∏ℓ
j=1 sij and s =
∏n
i=1. Then siei ∈ A. It follows
that (st)x =
∑ℓ
i=1(sei)xi ∈ AI. Thus x ∈ (AI)[p] since st ∈ R \ p. This
shows that (EI)[p] ⊆ (AI)[p]. Hence (AI)[p] = (EI)[p]. In particular, if we
take I = R, then we get A[p] = E[p].
(2) The containment (EI)[p] ⊆
(
EI[p]
)
[p]
is clear since EI ⊆ EI[p]. Let
x ∈
(
EI[p]
)
[p]
. Then tx ∈ EI[p] for some t ∈ R \ p. Thus tx =
∑k
i=1 viyi
with vi ∈ E and yi ∈ I[p] for 1 6 i 6 k. Let si ∈ R \ p such that siyi ∈ I,
and let s =
∏k
i=1 si. Then (st)x =
∑k
i=1 vi(syi) ∈ EI. It follows that
x ∈ (EI)[p]. Therefore, (EI)[p] = (EI[p])[p]
Theorem 2.6. Let R ⊆ S be a ring extension, and let I be a finitely
generated S-regular ideal of R. The following statements are equivalent.
(1) I is a quasi-cancellation ideal of the extension R ⊆ S.
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S. Tchamna 143
(2) For each prime ideal p of R, and for each S-regular finitely generated
R[p]-submodule E of S, we have [(EI)[p] : I[p]] = E[p].
Proof. (1) ⇒ (2) Suppose that I is a quasi-cancellation ideal of the
extension R ⊆ S, and let p be a prime ideal of R. Let E be a finitely
generated S-regular R[p]-submodule of S. Then E = (u1, . . . , uℓ)R[p] for
some elements u1, . . . , uℓ of S. Let A be the R-submodule of S generated
by u1, . . . , uℓ. Then by Proposition 2.3 and Proposition 2.4, we have
[AI : I] = A. It follows from [6, Proposition 2.1(4)] that [(AI)p : I[p]] = A[p].
Hence by Lemma 2.5, we have [(EI[p])[p] : I[p]] = [(EI)p : I[p]] = [(AI)p :
I[p]] = A[p] = E[p].
(2) ⇒ (1) Suppose that the statement (2) is true. Let A be an S-regular
finitely generated R-submodule of S, and let p be a prime ideal of R. Let
E = AR[p]. Then by Lemma 2.5, we have (AI)[p] = (EI)[p] and A[p] = E[p].
So, by hypothesis we have A[p] = E[p] = [(EI)[p] : I[p]] = [(AI)[p] : I[p]].
But by [6, Proposition 2.1(4)], we have [(AI)[p] : I[p]] = [(AI) : I][p].
Therefore, A[p] = [(AI) : I][p] for each prime ideal p of R. It follows from
Remark 1.1 that [AI : I] = A. Therefore, by the equivalence (1) ⇔ (2) of
Proposition 2.3, I is a quasi-cancellation ideal of the extension R ⊆ S.
In their book [5], Knebusch and Zhang defined the notion of Prüfer exten-
sion using valuation ring [5, Definition 1, p. 46]. Several characterizations
of a Prüfer extension are given in [5, Theorem 5.2, p. 47]. For the purpose
of this work, we will use the following: a ring extension R ⊆ S is called
Prüfer extension if R is integrally closed in S and R[α] = R[αn] for any
α ∈ S and any n ∈ N.
Lemma 2.7. [5, Theorem 1.13, p. 91] If a ring extension R ⊆ S is a
Prüfer extension, then every finitely generated S-regular R-submodule of
S is S-invertible.
Proposition 2.8. Let R ⊆ S be a ring extension, and let I be an S-regular
ideal of R.
(1) If I is a cancellation ideal of the extension R ⊆ S, then [I : I] = R.
(2) If the extension R ⊆ S is Prüfer, then the converse of statement (1)
is also true (i.e. in a Prüfer extension R ⊆ S, if I is an S-regular
ideal satisfying [I : I] = R, then I is a quasi-cancellation ideal).
Proof. (1) The proof follows directly from the equivalence (1) ⇔ (2) of
Theorem 2.3. It suffices to take J = R.
(2) Suppose that the extension R ⊆ S is Prüfer, and let I be an
S-regular ideal of R such that [I : I] = R. Let A be an S-regular finitely
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144 Cancellation ideals of a ring extension
generated R-submodule of S. Then by Lemma 2.7, A is S-invertible. We
show that A[I : I] = [AI : I]. Let x ∈ [AI : I]. Then xI ⊆ AI. Hence
xIA−1 ⊆ I. Thus xA−1 ⊆ [I : I]. It follows that x ∈ A[I : I]. On the
other hand, let y =
∑k
i=1 aivi ∈ A[I : I] with ai ∈ A and vi ∈ [I : I] for
1 6 i 6 k. Then viI ⊆ I. Hence aiviI ⊆ AI. Therefore, aivi ∈ [AI : I].
So y =
∑k
i=1 aivi ∈ [AI : I]. This shows that [AI : I] = A[I : I]. Hence
[AI : I] = A[I : I] = AR = A. Hence, by the equivalence (1) ⇔ (2) of
Proposition 2.3, I is a quasi-cancellation ideal of the extension R ⊆ S.
Let R ⊆ S be a ring extension. A nonzero S-regular ideal I of R is
called m-canonical ideal of the extension R ⊆ S if [I : [I : J ]] = J for all
S-regular ideal J of R. Properties of m-canonical ideals of a ring extension
are studied in [7].
Corollary 2.9. Any m-canonical ideal of a Prüfer extension is a quasi-
cancellation ideal.
Proof. If I is an m-canonical ideal of a Prüfer extension R ⊆ S, then by
[7, Proposition 2.3], we have [I : I] = R. It follows from Proposition 2.8(2)
that I is a quasi-cancellation ideal of the extension R ⊆ S.
Lemma 2.10. Let R ⊆ S be a ring extension, and let I be an S-regular
ideal of R which is a cancellation ideal of R ⊆ S. If I = (x, y) +A, where
A is an ideal of R containing mI for some maximal ideal m of R, then
I = (x) +A or I = (y) +A.
Proof. Let J = (x2 + y2, xy, xA, yA,A2)R. Then IJ = I3. Observe that
I2 is S-regular since I2S = I(IS) = IS = S. Also, from the equality
IJ = I3 we have (IJ)S = I3S = I(IS) = IS = S. So JS = S. This shows
that J is an S-regular ideal of R. It follows from the equation IJ = I3
and the fact that I is a cancellation ideal of the extension R ⊆ S that
J = I2. Thus x2 = t(x2 + y2) + terms from (xy, xA, yA,A2), with t ∈ R.
Suppose that t ∈ m. Then x2 ∈ (y2, xy, xA, yA,A2), since tx ∈ mI ∈ A.
Let K = (y)+A. Then I2 = IK. Furthermore, from the equality IK = I2,
we have K(IS) = I2S. Hence KS = S. Therefore, K is an S-regular ideal
of S. It follows that I = K since I is a cancellation ideal of the extension
R ⊆ S. The rest of the proof is similar to the proof of [2, Lemma].
Proposition 2.11. Let R ⊆ S be a ring extension, and let I be a nonzero
S-regular ideal of R. If I is a cancellation ideal of the extension R ⊆ S, then
for each maximal ideal m of R, there exists a ∈ R such that I[m] = (a)[m].
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S. Tchamna 145
Proof. Suppose that I is a cancellation ideal of the ring extension R ⊆ S,
and let m be a maximal ideal of R. Suppose that I ⊆ m. Then by
Lemma 2.10, and the proof of [2, Theorem], there exists a ∈ I such that
for each b ∈ I, (1 − u)b = ra for some u ∈ m and r ∈ R. Therefore,
(1− u)b ∈ (a). So b ∈ (a)[m]. This shows that I[m] ∈ (a)[m]. On the other
hand, the equality (a)[m] ⊆ I[m] is always true. Thus I[m] = (a)[m]. If I * m,
then I[m] = (1)[m] = R[m]. In fact, for x ∈ R[m], there exists s ∈ R \m such
that sx ∈ R. Thus (st)x ∈ I for each t ∈ I \m. It follows that x ∈ I[m].
Theorem 2.12. Let R ⊆ S be a ring extension, and let I be a nonzero
finitely generated S-regular ideal of R. The following statements are equiv-
alent.
(1) I is a cancellation ideal of the extension R ⊆ S.
(2) I is a quasi-cancellation ideal of the extension R ⊆ S.
(3) I is an S-invertible ideal of R.
(4) IR[X] is a cancellation ideal of the extension R[X] ⊆ S[X].
Proof. The equivalence (1) ⇔ (2) is the result of Theorem 2.4.
(1) ⇒ (3) Suppose that I is a cancellation ideal of the extension
R ⊆ S, and let m be a maximal ideal of R. By the previous proposition,
I[m] = (a)[m] for some a ∈ R. It follows that
(
I[m]
)
m[m]
=
(
(a)[m]
)
m[m]
. But
by [5, Lemma 2.9(b), p. 28], we have Im =
(
I[m]
)
m[m]
and (a)m =
(
(a)[m]
)
m[m]
.
Hence Im = (a)m. This shows that I is locally principal. It follows from
[5, Proposition 2.3, p. 97] that I is S-invertible.
(3) ⇒ (1) This implication is obvious.
(3) ⇒ (4) Suppose that I is an S-invertible ideal of the extension
R ⊆ S. First, note that (IR[X])(S[X]) = S[X] since IS = S. Hence
IR[X] is an S[X]-regular ideal of R[X]. Let J be the R-submodule of
S such that IJ = R. Then (IR[X])(JR[X]) = R[X]. This shows that
IR[X] is an S[X]-invertible ideal of R[X]. It follows from the equivalence
(1) ⇔ (3) that IR[X] is a cancellation ideal of the extension R[X] ⊆ S[X].
(4) ⇒ (1) Suppose that IR[X] is a cancellation ideal of the extension
R[X] ⊆ S[X]. Let J be an S-regular ideal of R. Then by the equivalence
(1) ⇔ (2) of Proposition 2.3, we have [(IR[X])(JR[X]) : IR[X]] = JR[X].
We show that [IJ : I] = J . First, note that the containment J ⊆ [IJ :
I] is always true. Let u ∈ [IJ : I]. Then uI ⊆ IJ . Therefore, uIR[X] ⊆
(IJ)R[X] ⊆ (IR[X])(JR[X]). Hence u ∈ [(IR[X])(JR[X]) : IR[X]] =
JR[X]. It follows that u ∈ JR[X] ∩ S = J . This shows that [IJ : I] ⊆
J . Hence [IJ : I] = J . It follows from the equivalence (1) ⇔ (2) of
Proposition 2.3 that I is a cancellation ideal of the extension R ⊆ S.
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146 Cancellation ideals of a ring extension
Corollary 2.13. Let R ⊆ S be a ring extension, and let I be a finitely
generated S-regular ideal of R. If I is a cancellation ideal of the extension
R ⊆ S, then I[m] is a cancellation ideal of the extension R[m] ⊆ S for each
maximal ideal m of R.
Proof. Let I be a finitely generated S-regular ideal of R, and let m be a
maximal ideal of R. Suppose that I is a cancellation ideal of the extension
R ⊆ S. Then by the previous theorem, I is S-invertible. Let J be an
R-submodule of S such that IJ = R. Then I[m]J[m] ⊆ (IJ)[m] ⊆ R[m].
Furthermore, since IS = S, there exist xi ∈ I and yi ∈ J , 1 6 i 6 ℓ, such
that 1 =
∑ℓ
i=1 xiyi. Let u ∈ R[m]. There exists t ∈ R \m such that tu ∈ R
and u =
∑ℓ
i=1(uxi)yi. But for 1 6 i 6 ℓ, t(uxi) = (tu)xi ∈ I since tu ∈ R
and xi ∈ I. It follows that uxi ∈ I[m]. Therefore, u =
∑ℓ
i=1(uxi)yi ∈
I[m]J ⊆ I[m]J[m]. This shows that R[m] ⊆ I[m]J[m]. Thus I[m]J[m] = R[m].
Hence I[m] is an S-invertible R[m]-submodule of S. It follows that I[m] is a
cancellation ideal of the extension R[m] ⊆ S, since an invertible ideal of
ring extension is always a cancellation ideal.
References
[1] Anderson, D.D., Anderson, D. F. (1984). Some remarks on cancellation ideals. Math
Japonica. 29 (6), pp 879-886.
[2] Anderson, D.D., Roitman, M. (1997). A characterization of cancellation ideals. Proc.
Amer. Math. Soc. No. 10, pp 2853 - 2854.
[3] Fuchs, L.; Salce, L. (2001) Modules over non-Noetherian domains. Mathematical
Surveys and Monographs, Vol. 84. American Mathematical Society, Providence, RI.
xvi+613 pp.
[4] Gabelli, S; Houston, E. (2000). Ideal theory in pullbacks. In Chapman S. T; Glaz
S., eds. Non-Noetherian commutative ring theory. Math. Appl., Vol 520. Dordrecht:
Kluwer Acad. Publ., pp. 199-227.
[5] Knebusch, M., Zhang, D. (2002). Manis valuations and Prüfer extensions I. Lecture
Notes in Mathematics, Vol. 1791. Berlin: Springer-Verlag.
[6] Paudel, L., Tchamna, S. (2018). On the saturation of submodules. Algebra and
Discrete Mathematics, Vol. 26, No. 1, pp. 110 - 123
[7] Tchamna, S. (2017). Multiplicative canonical ideals of ring extensions. Journal of
Algebra and Its Appl. Vol. 16, No. 4:170069.
Contact information
Simplice Tchamna Department of Mathematics
Georgia College, Milledgeville, GA, USA
E-Mail(s): simplice.tchamna@gcsu.edu
Received by the editors: 26.07.2019
and in final form 30.10.2020.
mailto:simplice.tchamna@gcsu.edu
S. Tchamna
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