Cyclic left and torsion-theoretic spectra of modules and their relations
In this paper, strongly-prime submodules of a cyclic module are considered and their main properties are given. On this basis, a concept of a cyclic spectrum of a module is introduced. This spectrum is a generalization of the Rosenberg spectrum of a noncommutative ring. In addition, some natural pro...
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irk-123456789-1551492019-06-17T01:27:18Z Cyclic left and torsion-theoretic spectra of modules and their relations Maloid-Glebova, M. In this paper, strongly-prime submodules of a cyclic module are considered and their main properties are given. On this basis, a concept of a cyclic spectrum of a module is introduced. This spectrum is a generalization of the Rosenberg spectrum of a noncommutative ring. In addition, some natural properties of this spectrum are investigated, in particular, its functoriality is proved. 2015 Article Cyclic left and torsion-theoretic spectra of modules and their relations / M. Maloid-Glebova // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 286-296. — Бібліогр.: 9 назв. — англ. 1726-3255 http://dspace.nbuv.gov.ua/handle/123456789/155149 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper, strongly-prime submodules of a cyclic module are considered and their main properties are given. On this basis, a concept of a cyclic spectrum of a module is introduced. This spectrum is a generalization of the Rosenberg spectrum of a noncommutative ring. In addition, some natural properties of this spectrum are investigated, in particular, its functoriality is proved. |
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Maloid-Glebova, M. |
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Maloid-Glebova, M. Cyclic left and torsion-theoretic spectra of modules and their relations Algebra and Discrete Mathematics |
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Maloid-Glebova, M. |
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Maloid-Glebova, M. |
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Cyclic left and torsion-theoretic spectra of modules and their relations |
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Cyclic left and torsion-theoretic spectra of modules and their relations |
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Cyclic left and torsion-theoretic spectra of modules and their relations |
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Cyclic left and torsion-theoretic spectra of modules and their relations |
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Cyclic left and torsion-theoretic spectra of modules and their relations |
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cyclic left and torsion-theoretic spectra of modules and their relations |
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Інститут прикладної математики і механіки НАН України |
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2015 |
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Cyclic left and torsion-theoretic spectra of modules and their relations / M. Maloid-Glebova // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 286-296. — Бібліогр.: 9 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT maloidglebovam cyclicleftandtorsiontheoreticspectraofmodulesandtheirrelations |
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2025-07-14T07:14:21Z |
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2025-07-14T07:14:21Z |
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1837605598015782912 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 20 (2015). Number 2, pp. 286–296
© Journal “Algebra and Discrete Mathematics”
Cyclic left and torsion-theoretic spectra
of modules and their relations
Marta Maloid-Glebova
Communicated by V. V. Kirichenko
Abstract. In this paper, strongly-prime submodules of a
cyclic module are considered and their main properties are given. On
this basis, a concept of a cyclic spectrum of a module is introduced.
This spectrum is a generalization of the Rosenberg spectrum of a
noncommutative ring. In addition, some natural properties of this
spectrum are investigated, in particular, its functoriality is proved.
Introduction
In this paper, we consider strongly-prime ideals and modules. The
concept of strongly-prime ideal was introduced by Beachy in [1]. Also
in that paper the author introduced and investigated the concept of a
strongly-prime module. Independently, the concept of strongly-prime
module and submodule were introduced and investigated by Dauns in his
paper [3]. Also, the strongly-prime modules were investigated by Algirdas
Kaučikas in [2], where the author studied strongly-prime submodules
of cyclic modules, but he did not study the concept of the Rosenberg
spectrum for modules. The concept of pre-order on ideals was introduced
by Rosenberg, and this concept is a basic one in the definition of cyclic
spectrum, whose functoriality is investigated in this paper. Also we con-
sider the notion and some properties of torsion-theoretic spectra of rings
and modules. The notion and main properties of torsion-theoretic spectra
were introduced by Golan in [5]. The main result of this paper is the
Key words and phrases: strongly-prime ideal, strongly-prime module, cyclic
spectrum, torsion-theoretic spectrum, localizations.
M. Maloid-Glebova 287
proof of the fact that there exists mapping from the cyclic spectrum to
the torsion-theoretic spectrum of module is continuous and surjective.
1. Strongly-prime ideals and modules
Let R be an associative ring with 1 6= 0. To have a reference, recall
some necessary concepts from the ring theory that are related to the
concept of spectrum of a noncommutative ring.
A left ideal p of a ring R is called prime, if for every x, y ∈ R, xRy ⊆ p
implies x ∈ p or y ∈ p. Clearly, any left prime ideal is two-sided if and
only if it is prime in the classical way. Set of all two-sided prime ideals is
denoted by Spec(R) and is called a (prime) spectrum of a ring R.
Recall the definition of a pre-order 6 on the set of left ideals of ring
R in the following way: a 6 b for left R-ideals a and b if and only if there
exists a finite subset V of ring R such that (a : V ) ⊆ b. A left prime
ideal p of a ring R is called a left Rosenberg point if (p : x) 6 p for any
x ∈ R\p, [8]. The set of all left Rosenberg points of a ring R is called a
left Rosenberg spectrum of R and is denoted by spec(R).
The space spec(R) may by defined in another way: this is the set of
all strongly prime left ideals. Recall that left ideal p of the ring R is called
strongly-prime, if for every x ∈ R\p there exist a finite set V of ring R
such that (p : V x) = {r ∈ R : rV x ⊆ p} ⊆ p. Clearly, every strongly-
prime left ideal of a ring R is a prime left ideal and every maximal
left ideal is strongly-prime. It is known that if R is noetherian, then
Spec(R) ⊆ spec(R).
Now let us recover the information about corresponding analogues of
the above concepts for left modules over a ring R.
The concept of strongly-prime module can be given in two ways.
A nonzero left module M over a ring R is called strongly-prime, if
for any nonzero x, y ∈ M there exists a finite subset {a1, a2, . . . , an} ⊆ R
such that AnnR{a1x, a2x, . . . , anx} ⊆ AnnR{y}, (ra1x = ra2x = · · · =
ranx = 0), r ∈ R implies ry = 0.
In [1], the authors introduced such a concept of strongly-prime sub-
module. A nonzero left module M over a ring R is called strongly-prime,
if for any nonzero x ∈ M there exists a finite subset {a1, a2, . . . , an} ⊆ R
such that AnnR{a1x, a2x, . . . , anx} = 0. If in this concept we put M = R,
we obtain the concept of a strongly-prime ring. Such strongly-prime rings
were studied in [4].
A submodule P of some module M is called strongly-prime, if the
quotient module M/P is a strongly-prime R-module. The set of all
288 Cyclic left and torsion-theoretic spectra
strongly-prime submodules of module M is called the left prime spectrum
of M and is denoted by spec(M). In particular, a left ideal p ⊂ R is called
strongly-prime if the quotient module R/p is a strongly-prime R-module.
In terms of elements, left ideal p ⊂ R is strongly-prime if for every u /∈ p
there exists such elements {a1, ..., an} ⊆ R and a natural number n = n(u)
such that ra1u, . . . , ranu ∈ p, r ∈ R implies r ∈ p.
2. Preorder on the set of modules and cyclic left spectrum
of module
It is easy to see that if R is a left noetherian ring and p ∈ Spec(R),
then R/p is a left noetherian prime ring. This implies that it is sufficient
to prove that in a left noetherian prime ring R zero ideal belongs to
spec(R). But taking into account the assumption that R is a prime Goldie
ring, for any 0 6= x ∈ R any two-sided ideal RxR is essential, thus there
exists a regular element a =
∑n
i=1 rixsi ∈ RxR (Using Goldie theorem).
Let V = {r1, . . . , rn} and y ∈ (0 : V x), then ya =
∑
yrixsi = 0. Since a
is regular, it follows that y = 0, hence 0 ∈ spec(R) indeed.
Clearly, it is necessary to demonstrate how to calculate prime left
ideals in an easy example. For this purpose we use the following example.
Example 1. Consider the matrix ring R = M2(k) over a (commutative)
field k.
It is well known that Spec(R) = {0}. Let L be a nonzero left R-ideal
and 0 6= r ∈ L. Since all nonzero left ideals of the ring R are maximal,
L = Rr. Multiplying r by the matrix units e11 and e12 resp., it easily
follows that we may assume r to be of the form r =
(
a b
0 0
)
, for some
nonzero string ( a b ) ∈ k2. One thus finds L = R
(
a b
0 0
)
= [a, b]k.
Moreover, [a, b]k = [a′, b′]k if and only if there exists such c ∈ k that
a = ca′ and b = cb′. Then spec(R) = {[a, b]k | a, b ∈ k} may be identified
with the projective line P 1
k (with "generic point" (0) = [0, 0]k). (See [6])
As in [8] we introduce a preorder 6 on the set of all left ideals by
putting K 6 L for a pair of left R-ideals L and K if and only if there
exists a finite subset V of the ring R such that (K : V ) ⊆ L.
Let us try to establish a preorder on the modules. Let R be a regular
module over itself with generator 1. Then M = R · 1 is a cyclic module.
Theorem 1. Every cyclic module is isomorphic to the quotient module
of a regular module by the annihilator of a generator R · m = R/ Ann(m),
where Ann(m) is the left annihilator of a generator m.
M. Maloid-Glebova 289
Consider some submodules of a cyclic module M which is presented as
Rm = R/ Ann(m) for the generator m. Let L, K be some submodules. We
can represent L = A/ Ann(m) and K = B/ Ann(m) for some left ideals
A and B of a ring R. Then we define L = A/ Ann(m) 6 K = B/ Ann(m)
if and only if A 6 B as the Rosenberg ideals. All properties are carried
out. Thus the spectrum of a cyclic module is the set of all ideals that are
in the spectrum of ring R.
It is well known that any module is the sum of its cyclic submodules.
Then the cyclic spectrum of a arbitrary module M is defined as the union
of all spectra of its cyclic submodules. The cyclic spectrum of module
M is denoted by Cspec(M). Then we can define L 6 K ⇐⇒ Cspec(L) ⊆
Cspec(K) for all submodules of the module M and obtain a preorder on
the family of such submodules.
Example 2. Let M = {( a
b ) |a, b ∈ k} be module of columns with height 2
over ring R = M2(k), where k is commutative field.
This module is cyclic with generator e = ( 1
0 ), that is, M = R × ( 1
0 ).
Then Ann(( 1
0 )) = {
(
0 b
0 d
)
|b, d ∈ k}, thus M/ Ann(( 1
0 )) ∼= {( a 0
c 0 ) |a, c ∈ k}.
The maximal submodule is {(( 1
0 ))}, hence cyclic spectrum consists of one
point.
Lemma 1. Let L and K be left cyclic R-modules. Then L 6 K if and
only if there exists a cyclic left R-module X, a monomorphism X Ln
and an epimorphism X ։ K. In other words, there exists a diagram
(L)n X ։ K.
Proof. Recall the definition of preorder for submodules of a cyclic module.
Let L, K be some submodules. We can represent L = A/ Ann(m) and
K = B/ Ann(m) for some left ideals A and B of the ring R. Then we
define L = A/ Ann(m) 6 K = B/ Ann(m) iff A 6 B as Rosenberg ideals.
Thus consider two cyclic modules L and K. They are fully represented by
their ideals A and B. Than if A 6 B by the definition, than there exists
a finite subset V ⊆ R, such that (A : V ) 6 B. Put V = {v1, . . . , vn} and
let X = R~v be a cyclic module, where ~v = {v1, . . . , vn} ∈ (L)n. Than we
have
(0 : ~v) = ∩n
i=1(A : vi) = (A : V ) ⊆ B,
which implies that there exists a surjection X ։ K.
On the other hand, assume that there exists a diagram (L)n α
X ։β K. Thus we can find such element x ∈ X, that β(x) = ~1. Put
290 Cyclic left and torsion-theoretic spectra
α(x) = (~v1, . . . , ~vn) ∈ (L)n, where (~v1, . . . , ~vn) ∈ A for some vi ∈ R.
Put V = {v1, . . . , vn} and than we have
(A : V ) = ∩n
i=1(A : vi) = (0 : ~v) = (0 : x) ⊆ B,
so A 6 B and L 6 K.
Usually from the preorder 6 we obtain an equivalence relation ∼
as follows: K ∼ L iff K 6 L and L 6 K. The equivalence class of the
submodule L will be denoted by [L].
Lemma 2 (11). If P is a strongly-prime module, then for any element
x ∈ M the following properties are equivalent:
(1) x /∈ P;
(2) (P : x) 6 P;
(3) (P : x) ∈ [P].
Lemma 3. Let M be cyclic module. If P ∈ Cspec(M) and if L and K
are submodules such that L ∩ K 6 P, then either L 6 P or K 6 P.
Proof. Let L � P and K � P and let L∩K 6 P. Thus, by the definition,
there exist ideals A, B and p of the ring R, such that L = A/ Ann(m),
K = B/ Ann(m) and P = p/ Ann(m). Then there exists a finite subset
V of the ring R, such that (A∩B : V ) ⊆ p. Since A � p, this implies that
(A : F ) * p for some finite subset F of the ring R. Thus, if we take F = V ,
we obtain the fact, that (A : V ) * p. Now, if x ∈ (A : V ) − p, then there
exists a finite set W ⊆ R with the property that (p : Wx) ⊆ p. Since
K � p, we have b � p, get fact that (B : F ) � p for any finite set F ⊆ R.
In particular, this holds for F = WxV , thus we can find an element
y ∈ (B : WxV ) − p. Finally, x ∈ (A : V ) implies that yWxV ⊆ B,
and y belongs to the set (B : WxV ). Certainly, yWxV ⊆ B, then
yWxV ⊆ A ∩ B and yWx ⊆ (A ∩ B : V ) ⊆ p. Thus, y ∈ (p : Wx) ⊆ p,
that contradicts to the fact, that y /∈ p.
Similarly
Lemma 4. If P ∈ Cspec(R) and if L and K are submodules such that
LK 6 P, then either L 6 P or K 6 P.
Recall the operation of multiplication of the submodules of cyclic
module R/c. Any submodule of cyclic module can be viewed as the
quotient-module of some left ideal by some other left ideal. Let we have
two such submodules L ∼= a/c and K ∼= b/c. Then L ·K = a/c ·b/c = ab/c.
M. Maloid-Glebova 291
Lemma 5. Let P and Q be strongly-prime submodules of the cyclic
module M . Then the following holds:
(1) If P ∼ Q, then P ∩ Q is a strongly-prime module and P ∼ P ∩ Q;
(2) If P ∩ Q is a strongly-prime module, then either P ⊆ Q or P ⊇ Q
or P ∼ Q.
Proof. Let P and Q be strongly-prime submodules of a cyclic module
M . Thus, for every submodule of a cyclic module there exist ideals
P = p/ Ann(m) and Q = q/ Ann(m), where P 6 Q if and only if p 6 q as
Rosenberg ideals. Similarly, we can formulate the definition of equivalence
relation. Thus let p ∼ q and x /∈ p∩ q. Let x /∈ p, thus there exists a finite
subset V ⊆ R, such that (p : V x) ⊆ p. If x /∈ q, then (q : Wx) ⊆ q for some
finite subset W of the ring R. Let U = V ∪ W , then (p ∩ q : Ux) ⊆ p ∩ q.
If x ∈ q, then (q : V x) = R, hence (p ∩ q : V x) ⊆ p. Since p ∼ q by the
assumption, p 6 q, and thus (p : U) ⊆ q for some finite subset U ⊆ R,
and since we may assume that 1 ∈ U , we obtain
(p ∩ q : UV x) = ((p ∩ q : V x) : U) ⊆ (p : U) ⊆ q.
Moreover, since V ⊆ UV , we also have
(p ∩ q : UV x) ⊆ (p ∩ q : V x) ⊆ p,
hence (p ∩ q : UV x) ⊆ p ∩ q, thus p ∩ q is a strongly prime ideal. Clearly
p ∩ q 6 p. On the other hand, since p 6 q, there exists a finite subset
V ⊆ R, with (p : V ) ⊆ q. We may obviously assume that 1 ∈ V , thus we
have (p : V ) ⊆ p. Hence (p : V ) ⊆ p ∩ q, so p 6 p ∩ q and p ∼ p ∩ q.
Let us now assume that p∩q is a strongly-prime ideal while p * q and
p + q. Sinc such a p * q there exists an element x ∈ p− q. Thus x /∈ p∩ q
and we may find a finite subset V ⊆ R such that (p ∩ q : V x) ⊆ p ∩ q.
Since (p : V x) = R, this yields (q : V x) ⊆ p ∩ q ⊆ p, hence p 6 q. By
symmetry p > q, and thus p ∼ q.
We easy obtain the following corollary:
Corollary 1. Let P1, . . . ,Pn be a finite family of strongly-prime modules,
such that P1 ∼ · · · ∼ Pn, then ∩n
i=1Pi is a strongly-prime module and
P1 ∼ ∩n
i=1Pi.
For any left module M , it’s submodule N is called strongly two-sided,
if left annihilator of every element of N is two-sided ideal. Clearly, new
292 Cyclic left and torsion-theoretic spectra
submodule is two-sided. Thus the set of such submodules is not empty,
because the zero submodule is strongly two-sided submodule. The sum of
all strongly two-sided submodules is called the bound of the submodule N .
In other words, the bound of the module is the largest submodule among
those that have two-sided left annihilators for all their elements. In the
case when M = N we are talking about the concept of a bound of the
module. As follows, the bound of the module M is the largest strongly
two-sided submodule of the module M . Denote the bound of a submodule
N by b(N), the bound of the module M by b(M).
Lemma 6. For every strongly-prime left submodule P of the module M
we have b(p) ∈ Cspec(M).
Proof. Let x, y ∈ M by elements, such that xRy ⊆ b(P). Assume that
y /∈ b(P). Then there exists such an element s ∈ R with ys /∈ P. For
every r ∈ R, (xr)R(ys) ⊆ (xRy)s ⊆ b(P)s ⊆ b(P) ⊆ P. Hence rx ∈ P.
Thus xR ⊆ b(P), which proves the assertion.
Lemma 7. If L 6 K are left R-modules, then b(L) ⊆ b(K). Conversely,
if R is a left noetherian fully-bounded ring, and if b(L) ⊆ b(K), then
L 6 K.
Proof. Since L 6 K, there exists a representation L = A/ Ann(m) and
K = B/ Ann(m) for some left ideals A and B of the ring R. Then A 6 B.
Thus there exist a finite subset V ⊆ R, that (A : V ) ⊆ B. Then for every
elements r ∈ b(L) and s ∈ R, we have rs ∈ A, therefor r ∈ (A : s). Thus
r ∈ (A : V ) = ∩s∈V (A : s). Since the former is contained in B, we have
b(L) ⊆ K, hence b(L) ⊆ b(K).
On the other hand, if R is a left noetherian fully-bounded ring, then
there exists a finite subset V = {v1, . . . , vn} ⊆ R such that b(L) =
∩n
i=1(A : vi) = (A : V ). Hence (A : V ) = b(A) ⊆ b(B) ⊆ B, and A 6 B,
therefore L 6 K.
Corollary 2. Let L and K be left modules such that L ∼ K, then b(L) =
b(K). Moreover if R is a left noetherian fully-bounded ring, then the
converse is also true.
3. Functoriality of cyclic spectrum of module
The cyclic spectrum construction can be regarded as a contravariant
functor from the category of modules to the category of sets,
CSpec: Mod → Set .
M. Maloid-Glebova 293
A contravariant functor CSpec is a rule assigning to each module M
over an associative ring R the set CSpec(M), the cyclic spectrum, i.e. the
set of submodules that are related in that spectrum, and to each module
homomorphism f : M1 → M2 the map of sets
Cspec(M1) → Cspec(M2),
P 7→ f−1(P ).
Consider the endomorphism ring E = End(M), and also consider
the center of that ring, denoted by C = {c ∈ E | cr = rc, ∀r ∈ E}.
Consider the construction of partial algebra over the ring C. It is the
set Q with a reflexive, symmetric binary relation ⊥ ⊆ Q × Q (called
commeasurability), partial addition and multiplication operations "+"
and "·", that are functions I → Q, a scalar multiplication operation
E × Q → Q, and elements 0, 1 ∈ C, such that the following axioms are
satisfied:
(1) for all q ∈ Q, a⊥0 and a⊥1;
(2) the relation ⊥ is preserved by the partial binary operations: for all
q1, q2, q3 ∈ Q, with qi⊥qj (1 6 i, j 6 3) and for all λ ∈ C, one has
(q1 + q2)⊥q3, (q1 · q2)⊥q3 and (λq1)⊥q2;
(3) if qi⊥qj for 1 6 i, j 6 3, then the values of all polynomials in q1, q2
and q3 form a commutative algebra.
Commeasurability subalgebra of a partial C-algebra Q is a subset
Z ⊆ Q consisting of pairwise commeasurable elements that is closed
under C-scalar multiplication and the partial binary operations of Q.
Given functors K : A → B and S : A → C, we recall that the (right)
Kan extension of S along K is a functor L : B → C with a natural
transformation ε : LK → S such that for any other functor F : B → C
with a natural transformation η : FK → S there is a unique natural
transformation δ : F → L, such that η = ε ◦ (δK).
Theorem 2. The functor Cspec: Modop → Set, with the identity
natural transformation Cspec |Comm Modop → CSpec is the Kan extension
of the functor Cspec: Comm Modop → Set along the embedding
Comm Modop ⊆ Modop.
Proof. Let F : Mod → Set be a contravariant functor with a
fixed natural transformation η : F |Comm Mod → Spec. Consider func-
tor C-Spec: Comm Mod → CSpec. We need to show that there
294 Cyclic left and torsion-theoretic spectra
is a unique natural transformation δ : F → CSpec, that induces
η : F |Comm Mod → CSpec upon a restriction to Comm Mod ⊆ Mod. To
construct it, fix ring R and module M over it. For every submodule
N ⊆ M over ring R the inclusion N ⊆ M given a morphisms of sets
F (M) → F (N), and η provides a morphisms ηN : F (N) → CSpec(N);
these compose to give morphisms F (M) → CSpec(N). By naturality of
the morphisms involved, these maps of F (M) collectively form a cone
over the diagram obtained for submodules of module. By the universal
property of limit, there exists a unique arrow making corresponding
diagram commutative for all N ⊆ M .
Defined morphisms δM form the components of a natural transfor-
mation δ : F → CSec. By construction, δ induces η when restricted
to Comm Mod. Uniquness of δ is guaranteed by the uniqueness of the
indicated arrow used to define δM above.
4. Localisations
Recall some definitions. By a torsion-theoretic spectrum we mean the
space of all prime torsion theories (or prime Gabriel filters of a main
ring) in the category of left R-modules with Zarisky topology. Recall
that prime torsion theory π ∈ R − tors is a torsion theory, for which
π = χ(R/I) for some critical ideal I of the ring R, where R − tors is
class of all torsion theories of the category R-mod and χ(R/I) is the
torsion theory, cogenerated by module E(R/I). If τ is torsion theory of
the category R-mod, then left R-module M is called torsion free module
if and only if there exist R from M into some member of τ . Class of all
torsion free modules for some τ is denote by Fτ . Further information
about the prime torsion theories can be fund in [5].
Remark 1. The class of all torsion theories R-tors can be partially
ordered by setting τ 6 τ ′ if and only if Tτ ⊆ Tτ ′ , namely, the class of
all torsion modules of one torsion theory is contained in the class of all
torsion modules of other torsion theory.
Introduce the notion of torsion-theoretic spectrum of a module M . Use
the concepts of torsion-theoretic spectrum of a ring R introduced above.
Introduce the concept of support of module M : supp(M) = {σ|σ(M) 6= 0}.
Torsion-theoretic spectrum of module M , R-Sp(M) is defined as R-sp(R)∩
supp(M).
If M is a left R-module, denote by ξ(M) the smallest torsion theory
such that M will be a torsion module, by χ(M) the largest torsion theory,
M. Maloid-Glebova 295
that M will be a torsion-free module. Clearly, Tχ(M) consists of R-modules
N such that HomR(N, E(M)) = 0, where E(M) is the injective hull of a
module M .
Lemma 8. If σ is a torsion theory and P is a left Rosenberg point of a
cyclic module M , then M/P is either a σ-torsion module or a σ-torsion
free module.
Proof. Assume that M/P /∈ Fσ. If P is a left Rosenberg point, then there
exists ideal p of a ring R such that P = p/ Ann(m). Pick an element
0 = x̄ ∈ σ(R/p). Thus, there exists a finite subset V of the ring R with
(p : V x) ⊆ p. Obviously, V x̄ ⊆ σ(R/p), hence, for every element v ∈ V
there exists left ideal Lv ∈ L(σ) such that Lvvx ⊆ p. Let L = ∩v∈LLv,
then L ∈ L(σ) and LV x ⊆ p. Hence L ⊆ (p : V x) ⊆ p and p ∈ L(σ), and
therefore M/P is σ-torsion module.
Proposition 1. If M is a fully bounded left noetherian module and
P ∈ Cspec(M), then the torsion theory τP = χ(M/P) cogenerated by
module M/P is prime.
Proof. Obviously, P /∈ L(τP), therefore M/P is a τP-torsion free module.
Thus, since χ(M/P) is the largest torsion theory for which M/P is
torsion free module. We have χ(M/P) 6 τP. Conversely, assume that
L(χ(M/P)) * L(τP). Take L ∈ L(χ(M/P)) − L(τP), then L 6 P. Thus,
by the definition, A 6 p for some ideals A and p of the ring R. Thus there
exists a finite subset U ⊆ R such that ∩u∈U (A : u) = (A : U) ⊆ p. Hence
p ∈ L(χ(M/P)), contradicting the definition of χ(M/P).
The previous statements imply the following result.
Theorem 3. The mapping Φ: Cspec(M)→M-sp, where Φ(P)=χ(M/P)
is continuous and surjective.
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Contact information
M. Maloid-Glebova Ivan Franko National University of L’viv
E-Mail(s): martamaloid@gmail.com
Received by the editors: 05.10.2015
and in final form 22.12.2015.
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