Annihilator-based dependency relations in modules and radical characterizations
In this paper we introduce and investigate annihilator based dependency relations for submodules of a unitary left \(R\)-module \(M\) over a commutative Noetherian ring \(R\). We show that two submodules \(N_1, N_2 \leq M\) are radically dependent (in the sense that \(\sqrt{\text{Ann}(N_1+N_2)}=\sqr...
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| author | Pekin, Ayten Özkaya, Hamdullah |
| author_facet | Pekin, Ayten Özkaya, Hamdullah |
| author_institution_txt_mv | [
{
"author": "Ayten Pekin",
"institution": "Istanbul University"
},
{
"author": "Hamdullah Özkaya",
"institution": "Bursa Technical University"
}
] |
| author_sort | Pekin, Ayten |
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| datestamp_date | 2026-07-08T07:55:33Z |
| description | In this paper we introduce and investigate annihilator based dependency relations for submodules of a unitary left \(R\)-module \(M\) over a commutative Noetherian ring \(R\). We show that two submodules \(N_1, N_2 \leq M\) are radically dependent (in the sense that \(\sqrt{\text{Ann}(N_1+N_2)}=\sqrt{\text{Ann}(N_1)}+\sqrt{\text{Ann}(N_2)}\)) if and only if \(\sqrt{\text{Ann}(N_1)}=\sqrt{\text{Ann}(N_2)}\). Building on this characterization, we introduce totally annihilator-dependent modules via a Krull-dimension condition and prove that, for a finitely generated module over a Noetherian ring, total annihilator-dependence is equivalent to \(\text{Ass}(M)\) being a singleton. We further study the Radical Distinction Set \(Z_g(M)\), establish its connection to associated primes, and extend the main results to finitely generated multiplication modules. |
| doi_str_mv | 10.12958/adm2471 |
| first_indexed | 2026-07-09T01:00:15Z |
| format | Article |
| fulltext |
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 41 (2026). Number 2, pp. 235–243
DOI:10.12958/adm2471
Annihilator-based dependency relations in
modules and radical characterizations
Ayten Pekin and Hamdullah Özkaya
Communicated by R. Wisbauer
Abstract. In this paper we introduce and investigate an-
nihilator based dependency relations for submodules of a unitary
left R-module M over a commutative Noetherian ring R. We
show that two submodules N1, N2 ≤ M are radically dependent
(in the sense that
√
Ann(N1 +N2) =
√
Ann(N1) +
√
Ann(N2))
if and only if
√
Ann(N1) =
√
Ann(N2). Building on this charac-
terization, we introduce totally annihilator-dependent modules via
a Krull-dimension condition and prove that, for a finitely genera-
ted module over a Noetherian ring, total annihilator-dependence
is equivalent to Ass(M) being a singleton. We further study the
Radical Distinction Set Zg(M), establish its connection to asso-
ciated primes, and extend the main results to finitely generated
multiplication modules.
Introduction
Classical submodule dependency in module theory is typically studied
through linear dependence or essential extensions. Such approaches,
however, do not capture the deeper interplay between annihilator ide-
als and the radical structure of a module. The annihilator Ann(N) of
a submodule N ≤ M encodes how N sits inside M from the ring’s per-
spective, while
√
Ann(N) records the prime ideal data governing the
2020 Mathematics Subject Classification: 13C05, 13C13, 13E05.
Key words and phrases: annihilator ideals, radical submodules, dependency
relations, coprimary modules, multiplication modules, associated primes.
https://doi.org/10.12958/adm2471
236 Annihilator-based dependency relations
support of N . When two submodules share the same radical annihila-
tor they behave identically at every prime, a constraint that turns out
to be both necessary and sufficient for what we call radical dependence.
This observation motivates a systematic study of modules in which all
pairs of submodules are radically dependent—the totally annihilator-
dependent modules—and their characterization via the set of associated
primes Ass(M).
A second thread of investigation concerns the Radical Distinction Set
Zg(M), which measures how far the pointwise annihilators of elements
of M deviate from the annihilator of the radical Rad(M). The vanishing
of Zg(M) provides a sufficient condition for total annihilator-dependence
and forces Ass(M) to be a singleton. Together, these concepts connect
prime ideal theory, Krull dimension, and the support-theoretic structure
of modules in a unified framework.
Throughout, R denotes a commutative ring with unity and M a uni-
tary left R-module. In Sections 2 and 2.4 we additionally assume that R
is Noetherian and M is finitely generated.
1. Definitions and basic concepts
1.1. Radical dependence
Definition 1 (Radically dependent submodules). Let N1, N2 ≤ M . We
say N1 and N2 are radically dependent if√
Ann(N1 +N2) =
√
Ann(N1) +
√
Ann(N2).
The following proposition shows that this condition admits a clean
equivalent formulation, which we use as the working characterization
throughout the paper.
Theorem 1 (Characterization of radical dependence). Let N1, N2 ≤ M .
Then N1 and N2 are radically dependent if and only if
√
Ann(N1) =√
Ann(N2).
Proof. (⇒) Suppose N1 and N2 are radically dependent, i.e.√
Ann(N1 +N2) =
√
Ann(N1) +
√
Ann(N2).
Since Ann(N1 + N2) = Ann(N1) ∩ Ann(N2), the left-hand side equals√
Ann(N1)∩
√
Ann(N2). Hence
√
Ann(N1)∩
√
Ann(N2) =
√
Ann(N1)+√
Ann(N2). For any two ideals I, J of R, I ∩ J = I + J forces I = J
(since I ⊆ I + J = I ∩ J ⊆ I). Therefore
√
Ann(N1) =
√
Ann(N2).
A. Pekin, H. Özkaya 237
(⇐) Suppose
√
Ann(N1) =
√
Ann(N2) =: p. Then√
Ann(N1 +N2) =
√
Ann(N1) ∩Ann(N2)
=
√
Ann(N1) ∩
√
Ann(N2)
= p ∩ p = p,
and
√
Ann(N1) +
√
Ann(N2) = p + p = p, so the radical dependence
condition holds.
Remark 1. Theorem 1 shows that the radical dependence relation is
symmetric (N1 and N2 are radically dependent ⇔ N2 and N1 are), and
that it partitions the set of nonzero submodules of M into classes of
equal radical annihilator. This is precisely the equivalence relation ∼γ
introduced in Section 1.4 below.
1.2. Totally annihilator-dependent modules
Definition 2 (Totally annihilator-dependent module). A finitely genera-
ted R-module M is called totally annihilator-dependent if for every pair
of nonzero submodules N1, N2 ≤ M satisfying Ann(Ni) ̸= 0,
dimR/Ann(N1 +N2) = min{dimR/Ann(N1), dimR/Ann(N2)}.
1.3. The radical distinction set
Definition 3 (Radical distinction set). Let M be an R-module. The
radical distinction set of M is
Zg(M) = {m ∈ M \ {0} | Ann(Rm) ̸= Ann(Rad(M))}.
A coarser variant, comparing radicals rather than annihilators directly,
is
Z̃g(M) = {m ∈ M \ {0} |
√
Ann(Rm) ̸=
√
Ann(Rad(M))}.
When Zg(M) = ∅, every nonzero element of M satisfies Ann(Rm)
= Ann(Rad(M)). Note that Z̃g(M) ⊆ Zg(M) in general, since exact
equality implies radical equality but not conversely.
When Rad(M)=0 (for instance whenM is semisimple),Ann(Rad(M))
= R, and every nonzero m satisfies Ann(Rm) ̸= R, so Zg(M) = M \{0}.
238 Annihilator-based dependency relations
1.4. Annihilator radical equivalence relation
Let m1,m2 ∈ M \ {0}. Define m1 ∼γ m2 if and only if
√
Ann(Rm1) =√
Ann(Rm2). This is an equivalence relation on M \{0}, and the equiva-
lence class of m is
[m]γ = {m′ ∈ M \ {0} |
√
Ann(Rm′) =
√
Ann(Rm)}.
This classification is finer than the partition induced by Zg(M) alone, as
it groups elements by their radical annihilator type. By Theorem 1, two
nonzero cyclic submodules Rm1 and Rm2 are radically dependent if and
only if m1 ∼γ m2.
2. Main results
Throughout this section R is a Noetherian commutative ring and M a
finitely generated R-module.
2.1. Characterization via associated primes
Theorem 2. Let R be a commutative Noetherian ring with identity, and
let M be a finitely generated R-module. If Ass(M) = {p}, then M is
totally annihilator-dependent.
Proof. Assume that Ass(M) = {p}. Let N1, N2 ≤ M be nonzero sub-
modules satisfying Ann(Ni) ̸= 0. Since M is finitely generated over a
Noetherian ring, every submodule of M is also finitely generated. More-
over, since R is a Noetherian ring, for 0M ̸= Ni ≤ M we have Ass(Ni) ̸= ∅
and Ass(Ni) ⊆ Ass(M) = {p}, and therefore necessarily Ass(Ni) = {p}.
This shows that p is the unique minimal prime over Ann(Ni). By Krull
dimension theory, the Krull dimension of a quotient ring is determined
by its minimal primes. Therefore√
Ann(Ni) =
⋂
q∈Ass(M)
q = q (i = 1, 2).
Hence
dim
R√
Ann(Ni)
= dim
R
Ann(Ni)
= dim(R/p) =: d
is obtained, since Krull dimension depends only on the radical of the
ideal.
A. Pekin, H. Özkaya 239
Now observe that
Ann(N1 +N2) = Ann(N1) ∩Ann(N2).
Indeed, r(N1 +N2) = 0 if and only if rN1 = 0 and rN2 = 0. Thus√
Ann(N1 +N2) =
√
Ann(N1) ∩Ann(N2) =
√
Ann(N1) ∩
√
Ann(N2)
is obtained. Since
√
Ann(Ni) = p, we get√
Ann(N1 +N2) = p ∩ p = p.
Consequently, again because Krull dimension depends only on the radi-
cal, we have
dim
R
Ann(N1 +N2)
= dim
R
p
= min{d, d},
and the desired equality is satisfied.
Theorem 3. Let R be a commutative Noetherian ring with identity, and
let M be a finitely generated R-module. Suppose that for p1, p2 ∈ Ass(M),
whenever p1 ̸= p2, one has
dim
R
p1
̸= dim
R
p2
.
If M is totally annihilator-dependent, then Ass(M) is a singleton.
Proof. Assume that M is totally annihilator-dependent. To obtain a
contradiction, suppose that Ass(M) contains two distinct prime ideals
p1, p2. By the definition of associated primes, for i = 1, 2, there exists
mi ∈ M such that
Ann(mi) = pi.
Let Ni := Rmi. Then
Ann(Ni) = Ann(mi) = pi
since R is commutative with identity, and hence√
Ann(Ni) = pi
is obtained. As above,
Ann(N1 +N2) = Ann(N1) ∩Ann(N2) = p1 ∩ p2.
240 Annihilator-based dependency relations
Therefore √
Ann(N1 +N2) =
√
p1 ∩ p2 = p1 ∩ p2
is obtained. Since
dim
R
p1 ∩ p2
= max
{
dim
R
p1
, dim
R
p2
}
,
and M is totally annihilator-dependent, we obtain
dim
R
p1
= dim
R
p2
.
This is a contradiction. Consequently, Ass(M) consists of a single prime
ideal.
2.2. The radical distinction set and singleton Ass
Theorem 4. Let R be a Noetherian commutative ring and M ̸= 0 a
finitely generated R-module. If Zg(M) = ∅, then
Ass(M) = {Ann(Rad(M))}
and Ann(Rad(M)) is a prime ideal.
Proof. Since M is nonzero and finitely generated over a Noetherian ring,
Ass(M) ̸= ∅. Let Ann(m) ∈ Ass(M) for some nonzero m. Since Rm
is cyclic, Ann(m) = Ann(Rm). Because Zg(M) = ∅, every nonzero m
satisfies Ann(Rm) = Ann(Rad(M)), hence Ann(m) = Ann(Rad(M)).
Since all associated primes coincide with Ann(Rad(M)), we conclude
Ass(M) = {Ann(Rad(M))}. As an element of Ass(M), this ideal is
prime.
Corollary 1. If Zg(M) = ∅, then M is totally annihilator-dependent.
2.3. Radical sum equivalence
Theorem 5. Let R be a Noetherian commutative ring and M a finitely
generated R-module. The following are equivalent:
(i) for all nonzero N1, N2 ≤ M :
√
Ann(N1 +N2) =
√
Ann(N1) +√
Ann(N2);
(ii) Ass(M) = {p} for some prime ideal p.
A. Pekin, H. Özkaya 241
Proof. (i) ⇒ (ii) Suppose the radical sum formula holds for all nonzero
N1, N2 ≤ M . Assume for contradiction that Ass(M) ⊇ {p1, p2} with
p1 ̸= p2. Choose Ni = Rmi with Ann(mi) = pi, so
√
Ann(Ni) = pi.
Then √
Ann(N1 +N2) =
√
Ann(N1) ∩Ann(N2) = p1 ∩ p2,
while
√
Ann(N1)+
√
Ann(N2) = p1+p2. Since p1 ̸= p2, we have p1∩p2 ⊊
p1+p2 (for if p1∩p2 = p1+p2 then, as in the proof of Theorem 1, p1 = p2,
a contradiction). Hence the radical sum formula fails for N1 and N2, a
contradiction. Therefore Ass(M) = {p}.
(ii) ⇒ (i) If Ass(M) = {p}, then for every nonzero N ≤ M , Ass(N) =
{p}, so
√
Ann(N) = p. By Theorem 1, any two nonzero submodules are
radically dependent, which gives√
Ann(N1 +N2) = p ∩ p = p = p+ p =
√
Ann(N1) +
√
Ann(N2).
2.4. Extension to multiplication modules
Multiplication modules were first introduced by Barnard [3] and further
developed by El-Bast and Smith [4]. For other references on multiplica-
tion modules, see [1].
Definition 4. An R-module M is a multiplication module if every sub-
module of M is of the form IM , for some ideal I of R.
Theorem 6. Let R be a Noetherian commutative ring and M a finitely
generated multiplication R-module with Ass(M) = {p}. Then for all
nonzero N1, N2 ≤ M ,√
Ann(N1 +N2) =
√
Ann(N1) +
√
Ann(N2) = p.
Proof. Since M is multiplication module, every submodule has the form
N = IM for some ideal I ≤ R, and Ann(IM) = {r ∈ R | rI ⊆ Ann(M)}.
For submodules N1 = I1M and N2 = I2M we have N1+N2 = (I1+I2)M ,
and the identity Ann(N1 + N2) = Ann(N1) ∩ Ann(N2) remains valid.
Since Ass(Ni) ⊆ Ass(M) = {p} and Ass(Ni) ̸= ∅, we get
√
Ann(Ni) = p,
and the conclusion follows by the argument in Theorem 5.
242 Annihilator-based dependency relations
3. Illustrative examples
Example 1 (M = Z/8Z). Let M = Z/8Z as a Z-module. Then Rad(M)
= 2Z/8Z = {0̄, 2̄, 4̄, 6̄} and Ann(Rad(M)) = 4Z. The annihilators of
nonzero elements are:
Ann(1̄) = Ann(3̄) = Ann(5̄) = Ann(7̄) = 8Z, Ann(2̄) = Ann(6̄) = 4Z,
and
Ann(4̄) = 2Z.
Since Ann(Rad(M)) = 4Z, the elements whose annihilator differs from
4Z are precisely 1̄, 3̄, 4̄, 5̄, 7̄, giving Zg(M) = {1̄, 3̄, 4̄, 5̄, 7̄}.
On the other hand,
√
Ann(M) =
√
0 = 2Z and
√
Ann(Rad(M)) =√
4Z = 2Z. For every nonzero m̄ ∈ M we have
√
Ann(m̄) = 2Z (since
the only prime dividing |m̄| in Z/8Z is 2), so Z̃g(M) = ∅. This example
illustrates that annihilator equality and radical annihilator equality yield
genuinely different structural decompositions.
Example 2. Let M = Z/5Z. Since M is simple, Rad(M) = 0 and
Ann(Rad(M)) = Z. For every nonzero m ∈ M , Rm = M , so
Ann(Rm) = 5Z ̸= Z = Ann(Rad(M)).
Hence Zg(M) = M \ {0̄} = {1̄, 2̄, 3̄, 4̄}. Similarly,
√
Ann(Rm) = 5Z ̸=
Z =
√
Ann(Rad(M)), so Z̃g(M) = M \ {0̄} as well. For simple modules
the two sets always coincide.
Example 3 (Positive case). Let M = Z/25Z as a Z-module. This is a
finitely generated multiplication Z-module with Ass(M) = {5Z}. Every
nonzero submodule N ≤ M satisfies
√
Ann(N) = 5Z, so√
Ann(N1 +N2) =
√
Ann(N1) +
√
Ann(N2) = 5Z
for all nonzero N1, N2 ≤ M , confirming Theorem 6.
Remark 2 (Necessity of hypotheses). The hypotheses of finite genera-
tion and multiplication in Theorem 6 cannot be dropped. Consider M =
Zp∞ (the Prüfer p-group), which is divisible but not finitely generated.
Let N1 = Z/pZ and N2 = pM = M . Then N1+N2 = M , Ann(N1) = pZ,
and Ann(M) = 0, giving√
Ann(N1 +N2) = 0 ̸= pZ+ 0 =
√
Ann(N1) +
√
Ann(N2).
Note that Zp∞ falls outside the standing assumption of finite generation;
its inclusion here is solely to illustrate why that assumption is essential.
A. Pekin, H. Özkaya 243
References
[1] Anderson, D.D., Arabaci, T., Tekir, Ü., Koç, S.: On S-multiplication modules.
Comm. Algebra 48(8), 3398–3407 (2020). https://doi.org/10.1080/00927872.20
20.1737873
[2] Khaksari, A., Sharif, H., Ershad, M.: On prime submodules of multiplication
modules. Int. J. Pure Appl. Math. 17(1), 41–49 (2004)
[3] Barnard, A.: Multiplication modules. J. Algebra 71(1), 174–178 (1981). https://
doi.org/10.1016/0021-8693(81)90112-5
[4] El-Bast, Z.A., Smith, P.F.: Multiplication modules. Comm. Algebra 16(4),
755–779 (1988). https://doi.org/10.1080/00927878808823601
[5] Hassanzadeh-Lelekaami, D., Roshan-Shekalgourabi, H.: Pseudo-prime submodu-
les of modules. Math. Reports 18(68), 591–608 (2016)
[6] Smith, P.F.: Radical submodules and uniform dimension of modules. Turkish J.
Math. 28(3), 255–270 (2004)
[7] Wisbauer, R.: Foundations of Module and Ring Theory. Gordon and Breach
Science Publishers, London (1991). https://doi.org/10.1201/9780203755532
[8] Lee, S., Moo, Y., Varmazyar, R.: Associated prime submodules of a multiplication
module. HonamMath. J. 39(2), 275–296 (2017). https://doi.org/10.5831/HMJ.20
17.39.2.275
[9] Atani, S.E., Ghaleh, S.K.G.: On multiplication modules. Int. Math. Forum
1(21-24), 1175–1180 (2006). http://dx.doi.org/10.12988/imf.2006.06096
[10] Koç, S.: On annihilator multiplication modules. Preprint at https://arxiv.org/
abs/2510.03791 (2025)
Contact information
A. Pekin Department of Mathematics, Istanbul
University, Istanbul, Turkey
E-Mail: a.pekin@istanbul.edu.tr
H. Özkaya Department of Mathematics, Bursa Technical
University, Bursa, Turkey
E-Mail: hamdullah.ozkaya@btu.edu.tr
Received by the editors: 18.02.2026
and in final form 11.06.2026.
https://doi.org/10.1080/00927872.2020.1737873
https://doi.org/10.1080/00927872.2020.1737873
https://doi.org/10.1016/0021-8693(81)90112-5
https://doi.org/10.1016/0021-8693(81)90112-5
https://doi.org/10.1080/00927878808823601
https://doi.org/10.1201/9780203755532
https://doi.org/10.5831/HMJ.2017.39.2.275
https://doi.org/10.5831/HMJ.2017.39.2.275
http://dx.doi.org/10.12988/imf.2006.06096
https://arxiv.org/abs/2510.03791
https://arxiv.org/abs/2510.03791
Ayten Pekin and Hamdullah Özkaya
|
| id | admjournalluguniveduua-article-2471 |
| institution | Algebra and Discrete Mathematics |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-09T01:00:15Z |
| publishDate | 2026 |
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| record_format | ojs |
| resource_txt_mv | admjournalluguniveduua/32/5f4113fd78e3212bc57b6b2da8c64532.pdf |
| spelling | admjournalluguniveduua-article-24712026-07-08T07:55:33Z Annihilator-based dependency relations in modules and radical characterizations Pekin, Ayten Özkaya, Hamdullah annihilator ideals, radical submodules, dependency relations, coprimary modules, multiplication modules, associated primes 13C05, 13C13, 13E05 In this paper we introduce and investigate annihilator based dependency relations for submodules of a unitary left \(R\)-module \(M\) over a commutative Noetherian ring \(R\). We show that two submodules \(N_1, N_2 \leq M\) are radically dependent (in the sense that \(\sqrt{\text{Ann}(N_1+N_2)}=\sqrt{\text{Ann}(N_1)}+\sqrt{\text{Ann}(N_2)}\)) if and only if \(\sqrt{\text{Ann}(N_1)}=\sqrt{\text{Ann}(N_2)}\). Building on this characterization, we introduce totally annihilator-dependent modules via a Krull-dimension condition and prove that, for a finitely generated module over a Noetherian ring, total annihilator-dependence is equivalent to \(\text{Ass}(M)\) being a singleton. We further study the Radical Distinction Set \(Z_g(M)\), establish its connection to associated primes, and extend the main results to finitely generated multiplication modules. Lugansk National Taras Shevchenko University 2026-07-08 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2471 10.12958/adm2471 Algebra and Discrete Mathematics; Vol 41, No 2 (2026) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2471/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2471/1372 Copyright (c) 2026 Algebra and Discrete Mathematics |
| spellingShingle | annihilator ideals radical submodules dependency relations coprimary modules multiplication modules associated primes 13C05 13C13 13E05 Pekin, Ayten Özkaya, Hamdullah Annihilator-based dependency relations in modules and radical characterizations |
| title | Annihilator-based dependency relations in modules and radical characterizations |
| title_full | Annihilator-based dependency relations in modules and radical characterizations |
| title_fullStr | Annihilator-based dependency relations in modules and radical characterizations |
| title_full_unstemmed | Annihilator-based dependency relations in modules and radical characterizations |
| title_short | Annihilator-based dependency relations in modules and radical characterizations |
| title_sort | annihilator-based dependency relations in modules and radical characterizations |
| topic | annihilator ideals radical submodules dependency relations coprimary modules multiplication modules associated primes 13C05 13C13 13E05 |
| topic_facet | annihilator ideals radical submodules dependency relations coprimary modules multiplication modules associated primes 13C05 13C13 13E05 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2471 |
| work_keys_str_mv | AT pekinayten annihilatorbaseddependencyrelationsinmodulesandradicalcharacterizations AT ozkayahamdullah annihilatorbaseddependencyrelationsinmodulesandradicalcharacterizations |