An outer measure on a commutative ring
We show how to produce a reasonable outer measure on a commutative ring from a given measure on a family of prime ideals of this ring. We provide a few examples and prove several properties of such outer measures.
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| citation_txt | An outer measure on a commutative ring / D. Dudzik, M. Skrzyński // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 51-58. — Бібліогр.: 3 назв. — англ. |
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| description | We show how to produce a reasonable outer measure on a commutative ring from a given measure on a family of prime ideals of this ring. We provide a few examples and prove several properties of such outer measures.
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 21 (2016). Number 1, pp. 51–58
© Journal “Algebra and Discrete Mathematics”
An outer measure on a commutative ring
Dariusz Dudzik and Marcin Skrzyński
Communicated by V. I. Sushchansky
Abstract. We show how to produce a reasonable outer
measure on a commutative ring from a given measure on a family
of prime ideals of this ring. We provide a few examples and prove
several properties of such outer measures.
Introduction
Throughout the present paper, R is a nonzero commutative ring with
identity. We denote by Spec(R) the family of all the prime ideals of R.
(Notice that, by definition, every prime ideal is proper).
It is well known [1] that topological properties of Spec(R) equipped
with the Zariski topology reflect algebraic properties of R. But are there
useful relationships between algebraic or geometric properties of R and
measures on Spec(R)? This question seems to be quite interesting and
not worked out in the specialist literature. The present paper provides
some basic remarks concerning the question and, hopefully, is a starting
point for further study.
In the paper, we will show that an arbitrary measure on a suitable
subfamily of Spec(R) induces an outer measure on R with good multi-
plicative properties. We will also discuss a few elementary examples of
such outer measures.
2010 MSC: 13A15, 28A12.
Key words and phrases: outer measure, measure, commutative ring, prime
ideal.
52 An outer measure on a commutative ring
By “measure” we mean a “non-negative σ-additive measure”. We
denote by 2X the power set of a set X. We define
|X| =
{
the cardinality of X, if X is finite,
+∞, otherwise.
By R× we denote the set of invertible elements of R. Notice that
℘ ∩ R× = ∅ whenever ℘ ∈ Spec(R). We define Max(R) to be the family
of all the maximal ideals of R. One can prove that Max(R) ⊆ Spec(R)
and
⋃
Max(R) = R \ R×.
We refer to [1] for more information about commutative rings and
to [2] for elements of measure theory.
1. Construction
We will use the definition of outer measure taken from [2].
Definition 1. We say that µ∗ : 2X −→ [0, +∞] is an outer measure on
a set X, if the following conditions are satisfied:
(1) µ∗(A) 6
∞∑
n=1
µ∗(Bn) for every sequence {Bn}∞
n=1 of subsets of X
and every A ⊆
∞⋃
n=1
Bn,
(2) µ∗(∅) = 0.
Let P ⊆ Spec(R) be such that
⋃
P = R\R×, and let M be a σ-algebra
of subsets of P. For a set A ⊆ R we define
Ω(A) =
{
S ∈ M :
⋃
S ⊇ A \ R×
}
.
Proposition 1. Suppose that µ : M −→ [0, +∞] is a measure. Then the
function µ∗ : 2R −→ [0, +∞] defined by
µ∗(A) = inf
S∈Ω(A)
µ(S)
is an outer measure on R. (This outer measure will be referred to as the
outer measure induced by µ).
Proof. It is obvious that µ∗(∅) = 0. Let {Bn}∞
n=1 be a sequence of subsets
of R and let ε be an arbitrary positive real number. Observe that
∀ n ∈ N \ {0} ∃ Sn ∈ Ω(Bn) : µ(Sn) 6 µ∗(Bn) +
ε
2n
.
D. Dudzik, M. Skrzyński 53
If A ⊆
∞⋃
n=1
Bn, then
∞⋃
n=1
Sn ∈ Ω(A) and hence
µ∗(A) 6
∞∑
n=1
µ(Sn) 6 ε +
∞∑
n=1
µ∗(Bn).
Since ε is arbitrary, the above inequalities yield µ∗(A) 6
∞∑
n=1
µ∗(Bn).
The outer measure induced by a measure on a family of prime ideals
is a slight modification of a well known measure-theoretical construction.
In the next section we give examples that illustrate and motivate this
modification.
2. Examples
We denote by (a) the principal ideal generated by an element a ∈ R.
Consider a further example of a “covering by prime ideals”.
Example 1. We assume that R is a unique factorization domain and
define Pirr(R) = {(0)} ∪ {(a) : a ∈ R, a is irreducible}. Observe that
Pirr(R) ⊆ Spec(R) and
⋃
Pirr(R) = R \ R×. Moreover, if n ∈ N \ {0, 1}
and R = C[x1, . . . , xn], then Pirr(R) ∩ Max(R) = ∅.
Recall that for every ideal I of the ring of integers there exists exactly
one m ∈ N ∪ {0} such that I = (m). Notice also that Max(Z) = {(p) :
p ∈ P}, where P stands for the set of prime numbers.
Proposition 2. Let µ∗ : 2Z −→ [0, +∞] be the outer measure in-
duced by the counting measure on Max(Z), and let A ⊆ Z be such that
A \ {−1, 1} 6= ∅. Then
(i) µ∗({−1, 1}) = 0,
(ii) µ∗(A) = 1 if and only if
∃ d ∈ N \ {0, 1} ∀ k ∈ A \ {−1, 1} : d | k
(in particular, µ∗(A) = 1 whenever A is a singleton or a proper
ideal of Z),
(iii) µ∗(A) 6 |A|.
Moreover, in the case where A ∩ {−1, 1} = ∅ and A is a finite set,
µ∗(A) = |A| if and only if the elements of A are pairwise relatively prime.
54 An outer measure on a commutative ring
Proof. Since {−1, 1} = Z
×, we have ∅ ∈ Ω({−1, 1}). Equality (i) follows.
By the above characterization of Max(Z) and the definition of counting
measure, µ∗(A) = 1 if and only if A \ {−1, 1} ⊆ (p1) for a prime number
p1. The latter condition means precisely that
∃ p1 ∈ P ∀ k ∈ A \ {−1, 1} : p1 | k.
Finally, if d ∈ N \ {0, 1}, k ∈ Z and d | k, then k is divisible by every
prime factor of d. Property (ii) follows.
Property (iii) is an immediate consequence of the definition of outer
measure and the fact that µ∗({k}) 6 1 for all k ∈ Z.
Assume that A ∩ {−1, 1} = ∅ and A is a finite set. Let us define
ℓ = |A|. Observe that µ∗(A) 6= |A| if and only if
∃ S ∈ Ω(A) : |S| 6 ℓ − 1.
Since the cardinality of A is greater than the cardinality of S, the latter
condition holds true if and only if
∃ s, t ∈ A ∃ p2 ∈ P :
{
s 6= t,
s, t ∈ (p2),
and this means precisely that there exist two distinct elements of A which
are not relatively prime.
Let n ∈ N \ {0}. Consider a σ-algebra N of subsets of Cn, a measure
λ : N −→ [0, +∞], and the map
Φ : Cn ∋ z 7−→ {f ∈ C[x1, . . . , xn] : f(z) = 0} ∈ Max(C[x1, . . . , xn]).
The family M = {S ⊆ Max(C[x1, . . . , xn]) : Φ−1(S) ∈ N} is a σ-algebra
of subsets of Max(C[x1, . . . , xn]). The function η : M ∋ S 7→ λ(Φ−1(S)) ∈
[0, +∞] is a measure.
Let us define U = C[x1, . . . , xn]×. (Obviously, U = C \ {0}).
Proposition 3. If η∗ : 2C[x1,...,xn] −→ [0, +∞] is the outer measure
induced by η and A ⊆ C[x1, . . . , xn] is such that A \ U 6= {0}, then
η∗(A) = inf{λ(Z) : Z ∈ N, Z ∩ f−1(0) 6= ∅ for every f ∈ A \ C}.
Proof. If A ⊆ U , then {Z ∈ N : Z ∩ f−1(0) 6= ∅ for every f ∈ A \ C} =
N, and hence
inf{λ(Z) : Z ∈ N, Z ∩ f−1(0) 6= ∅ for every f ∈ A \ C} = 0 = η∗(A).
D. Dudzik, M. Skrzyński 55
By Hilbert’s Nullstellensatz, the map Φ is bijective. Consequently, M =
{Φ(Z) : Z ∈ N}. Suppose that A \ C 6= ∅. Then for any Z ∈ N the
following equivalences hold true:
Φ(Z) ∈ Ω(A) ⇐⇒ (∀ f ∈ A \ U ∃ ℘ ∈ Φ(Z) : f ∈ ℘) ⇐⇒
(∀ f ∈ A \ C ∃ z ∈ Z : f(z) = 0) ⇐⇒
(
∀ f ∈ A \ C : Z ∩ f−1(0) 6= ∅
)
.
(The second equivalence holds because 0 belongs to every ideal). Therefore,
η∗(A) = inf
S∈Ω(A)
η(S) =
= inf{λ(Φ−1(Φ(Z))) : Z ∈ N, Z ∩ f−1(0) 6= ∅ for every f ∈ A \ C},
which completes the proof.
Example 2. Let η∗ : 2C[x,y] −→ [0, +∞] be the outer measure induced by
the counting measure on Max(C[x, y]). Consider the set E = {f, g, h, k} ⊂
C[x, y], where
f(x, y) = x2 − y + 1, g(x, y) = y2, h(x, y) = xy − 1, k(x, y) = xy + 1.
Since f−1(0) ∩ g−1(0) ∩ h−1(0) ∩ k−1(0) = ∅ and f−1(0) ∩ g−1(0) 6= ∅,
we have η∗(E) ∈ {2, 3}. Observe that {f, g}, {f, h} and {f, k} are the
only two-element subsets of E which have a common zero. Consequently,
no three-element subset of E has a common zero. It follows, therefore,
that η∗(E) = 3.
Notice that in the example above, if I is a proper ideal of C[x, y], then
I ⊆ ℘ for an ideal ℘ ∈ Max(C[x, y]) and hence η∗(I) = 1.
Let K be a nonempty compact subset of Rn and let C(K,R) stand
for the ring of all the continuous functions f : K −→ R. Recall that
C(K,R)× = {f ∈ C(K,R) : f(x) 6= 0 for all x ∈ K}. The map
Ψ : K ∋ x 7−→ {f ∈ C(K,R) : f(x) = 0} ∈ Max(C(K,R))
is well known to be a bijection [3]. Consequently, if B is a σ-algebra of
subsets of K and ξ : B −→ [0, +∞] is a measure, then M = {Ψ(Z) : Z ∈
B} is a σ-algebra of subsets of Max(C(K,R)) and
η : M ∋ S 7→ ξ(Ψ−1(S)) ∈ [0, +∞]
is a measure. The obvious counterpart of Proposition 3 remains true.
56 An outer measure on a commutative ring
Example 3. Let η∗ : 2C(K,R) −→ [0, +∞] be the outer measure induced by
η. We will denote by W the set of all the polynomial functions f : K −→ R.
Since
∀ x ∈ K ∃ f ∈ W : f−1(0) = {x},
we have η∗(W ) = η∗(C(K,R)) = ξ(K).
Now, suppose that K is the Euclidean closed unit ball and ξ is the
n-dimensional Lebesgue measure. If E stands for the set of all the radially
symmetric functions belonging to C(K,R) and L is the straight line
segment that joins the origin to a boundary point of K, then
∀ f ∈ E \ C(K,R)× : L ∩ f−1(0) 6= ∅.
Consequently, η∗(E) = ξ(L) = 0 whenever n > 2. It is easy to see that if
n = 1, then η∗(E) = 1.
3. General properties
In the theorem below (it is the main result of the paper) we use
the notations and assumptions of Proposition 1. For n ∈ N \ {0} and
A1, . . . , An ⊆ R we define A1 . . . An = {a1 . . . an : a1 ∈ A1, . . . , an ∈ An}.
Moreover, if A ⊆ R, then An = {an : a ∈ A} and A•n = A . . . A
︸ ︷︷ ︸
n
.
Theorem 1. Let A, B ⊆ R and let C be a nonempty subset of R×. Then
(i) µ∗(R×) = 0,
(ii) µ∗(A) = µ∗(A \ R×),
(iii) µ∗({0}) = min{µ∗(E) : E ⊆ R, E \ R× 6= ∅},
(iv) ∀ n ∈ N \ {0} : µ∗(An) = µ∗(A•n) = µ∗(A),
(v) µ∗(AC) = µ∗(A),
(vi) µ∗(AB) > max{µ∗(A), µ∗(B)} whenever A ∩ R× 6= ∅ and
B ∩ R× 6= ∅,
(vii) µ∗(AB) 6 µ∗(A) + µ∗(B),
(viii) µ∗(AB) = µ∗(A) whenever A ∩ R× = ∅ and B ∩ R× 6= ∅,
(ix) µ∗(AB) = min{µ∗(A), µ∗(B)} whenever A ∩ R× = ∅ and
B ∩ R× = ∅.
Proof. Properties (i) and (ii) are obvious.
Property (iii) follows from the facts that 0 /∈ R× and 0 belongs to
every ideal of R.
Fix a positive integer n. Let a1, . . . , an ∈ R. The product a1 . . . an is
not invertible if and only if there exists an index i ∈ {1, . . . , n} such that
D. Dudzik, M. Skrzyński 57
ai is not invertible. Similarly, a1 . . . an ∈ ℘ for an ideal ℘ ∈ Spec(R) if and
only if there exists an index i ∈ {1, . . . , n} such that ai ∈ ℘. Therefore,
Ω(An) = Ω(A•n) = Ω(A). Property (iv) follows.
Let a ∈ R and c ∈ R×. Observe that ac /∈ R× if and only if a /∈ R×.
Moreover,
∀ ℘ ∈ Spec(R) : ac ∈ ℘ ⇔ a ∈ ℘.
Consequently, Ω(AC) = Ω(A).
Suppose that C1 = A ∩ R× 6= ∅ and C2 = B ∩ R× 6= ∅. Since
AC2 ∪BC1 ⊆ AB, we have max{µ∗(AC2), µ∗(BC1)} 6 µ∗(AB). Property
(v) yields µ∗(AC2) = µ∗(A) and µ∗(BC1) = µ∗(B). This completes the
proof of (vi).
Let S ∈ Ω(A) and T ∈ Ω(B). Suppose that ab /∈ R× for some a ∈ A
and b ∈ B. Then a /∈ R× or b /∈ R×. By the definition of ideal, we get
therefore
ab ∈
⋃
S ∪
⋃
T .
Consequently, S ∪ T ∈ Ω(AB) and hence µ∗(AB) 6 µ(S) + µ(T ). Since
S and T are arbitrarily chosen, it follows that µ∗(AB) 6 µ∗(A) + µ∗(B).
Assume that A ∩ R× = ∅ and C2 = B ∩ R× 6= ∅. Then, by the defi-
nition of ideal, Ω(A) ⊆ Ω(AB) which implies that µ∗(AB) 6 µ∗(A). On
the other hand, by (v), we have µ∗(A) = µ∗(AC2) 6 µ∗(AB). Therefore,
µ∗(AB) = µ∗(A).
Finally, assume that A ∩ R× = ∅ and B ∩ R× = ∅. Then µ∗(AB) 6
min{µ∗(A), µ∗(B)} (cf. the proof of property (viii)). Suppose now that
µ∗(AB) < min{µ∗(A), µ∗(B)}. Then
∃ U ∈ Ω(AB) :
{
µ∗(
⋃
U) < µ∗(A),
µ∗(
⋃
U) < µ∗(B).
(Notice that µ∗(
⋃
U) 6 µ(U)). Consequently,
µ∗(A \
⋃
U) > µ∗(A) − µ∗(A ∩
⋃
U) > µ∗(A) − µ∗(
⋃
U) > 0
and, in the same way, µ∗(B \
⋃
U) > 0. Since AB ∩ R× = ∅ and therefore
AB ⊆
⋃
U , we get
∃ a ∈ A ∃ b ∈ B ∃ ℘ ∈ U ⊆ Spec(R) :
{
ab ∈ ℘,
a /∈ ℘, b /∈ ℘,
a contradiction. Property (ix) follows.
58 An outer measure on a commutative ring
We will conclude the paper with an example illustrating the behavior
of µ∗(AB) in the case where A and B both contain invertible elements.
Example 4. Let µ∗ : 2Z −→ [0, +∞] be the outer measure induced by
the counting measure on Max(Z). If A = {1, 2, 3}, B1 = {1, 2, 3, 5}, B2 =
{1, 2, 5, 7} and B3 = {1, 5, 7, 11}, then µ∗(A) = 2, µ∗(B1) = µ∗(B2) =
µ∗(B3) = 3, µ∗(AB1) = 3, µ∗(AB2) = 4 and µ∗(AB3) = 5.
References
[1] M. F. Atiyah, I. G. MacDonald, Introduction to Commutative Algebra, Westview
Press, 1994.
[2] H. Federer, Geometric Measure Theory, Springer, 1969.
[3] W. Rudin, Functional Analysis, McGraw-Hill, 1991.
Contact information
D. Dudzik Institute of Mathematics,
Pedagogical University of Cracow,
ul. Podchora̧żych 2,
30-084 Kraków, Poland
E-Mail(s): dariusz.dudzik@gmail.com
M. Skrzyński Institute of Mathematics,
Cracow University of Technology,
ul. Warszawska 24,
31-155 Kraków, Poland
E-Mail(s): pfskrzyn@cyf-kr.edu.pl
Received by the editors: 15.10.2015
and in final form 05.01.2016.
|
| id | nasplib_isofts_kiev_ua-123456789-155199 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-24T12:54:40Z |
| publishDate | 2016 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Dudzik, D. Skrzyński, M. 2019-06-16T10:48:33Z 2019-06-16T10:48:33Z 2016 An outer measure on a commutative ring / D. Dudzik, M. Skrzyński // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 51-58. — Бібліогр.: 3 назв. — англ. 1726-3255 2010 MSC:13A15, 28A12. https://nasplib.isofts.kiev.ua/handle/123456789/155199 We show how to produce a reasonable outer measure on a commutative ring from a given measure on a family of prime ideals of this ring. We provide a few examples and prove several properties of such outer measures. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics An outer measure on a commutative ring Article published earlier |
| spellingShingle | An outer measure on a commutative ring Dudzik, D. Skrzyński, M. |
| title | An outer measure on a commutative ring |
| title_full | An outer measure on a commutative ring |
| title_fullStr | An outer measure on a commutative ring |
| title_full_unstemmed | An outer measure on a commutative ring |
| title_short | An outer measure on a commutative ring |
| title_sort | outer measure on a commutative ring |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155199 |
| work_keys_str_mv | AT dudzikd anoutermeasureonacommutativering AT skrzynskim anoutermeasureonacommutativering AT dudzikd outermeasureonacommutativering AT skrzynskim outermeasureonacommutativering |