Ideal operator

In abstract algebra , an ideal is a subset of an algebraic structure with at least one multiplicative two-digit operation that is closed with respect to products with elements from the entire structure.

The ideals of the same type on a given algebraic structure always form a system of envelopes called the ideal system . For every ideal system there is always a corresponding envelope operator (and vice versa), that is the corresponding ideal operator .

For the sake of simplicity, only the commutative case is described here. If one waives the commutativity of the multiplication, then the following are left ideals , and if the left and right factors are swapped for each product , right ideals result accordingly . Bilateral ideals, or just ideals, are both left and right ideals. In commutativity, there is no difference between these three types of ideals.

Ring ideals

Number-theoretical investigations of number ranges , in which a clear prime factorization of elements was no longer given, led to the development of the "classical" ideal theory for commutative rings .

definition

If a ring, then a ( Dedekind's ) ideal or ideal is the carrier set of a subgroup of for which: ${\displaystyle (R,+,\cdot )}$ ${\displaystyle d}$ ${\displaystyle A_{d}\subseteq R}$ ${\displaystyle (R,+)}$

${\displaystyle \forall r\in R\;\forall a\in A_{d}:}$

${\displaystyle r\cdot a\in A_{d}.}$

properties

• The ideals of a ring are precisely the kernels of the ring homomorphisms of the ring.
• The ideals of a ring each form an envelope system , so that the ideals are given by the associated envelope operator . ${\displaystyle (\;)_{d}}$

Remarks

• There were other ideal terms for rings, but also for other algebraic structures such as associations , semigroups , half rings, etc., which have (at least) one associative two-digit operation .
• There are also ideals in algebraic structures with non-associative two-digit operations, such as Lie algebras .
• The concept of the association ideal was also generalized to the order ideal for any semi-ordered sets .
• As a rule, the index is left out if it is clear which shell operator it is.

General ideal operators

Since, as a rule, only the respective associative two-digit operation is decisive for the factorization (the non-associative case is not dealt with below), it is sufficient for a general ideal theory to consider semigroups:

In the following, a commutative multiplicative semigroup is always given , and it is ${\displaystyle (S,\cdot )}$

${\displaystyle \cdot :{\mathfrak {P}}(S)\times {\mathfrak {P}}(S)\to {\mathfrak {P}}(S),(A,B)\mapsto A\cdot B:=\{a\cdot b\mid a\in A,b\in B\},}$

the complex multiplication over , where is the power set of . ${\displaystyle (S,\cdot )}$ ${\displaystyle {\mathfrak {P}}(S):=\{A\mid A\subseteq S\}}$ ${\displaystyle S}$

${\displaystyle {\bigl (}{\mathfrak {P}}(S),\cup ,\cap ,\cdot {\bigr )}}$then forms a commutative, associative, complete multiplicative lattice with a zero element . ${\displaystyle \emptyset }$

definition

It should now

${\displaystyle (\;)_{x^{*}}:{\mathfrak {P}}(S)\to {\mathfrak {P}}(S),A\mapsto (A)_{x^{*}},}$

be an envelope operator on , with the property that ${\displaystyle S}$

${\displaystyle \forall A\in {\mathfrak {P}}(S):}$

${\displaystyle S\cdot (A)_{x^{*}}\subseteq (A)_{x^{*}}.}$

${\displaystyle (\;)_{x^{*}}}$Then, a -Idealoperator or shortly operator on called, which is -Idealsystem or system to a means -ideal and is of produced -Great . denotes of generated -ideal and is of generated - principal ideal . ${\displaystyle x^{*}}$ ${\displaystyle x^{*}}$ ${\displaystyle (S,\cdot )}$ ${\displaystyle {\mathfrak {I}}_{x^{*}}:=\{(A)_{x^{*}}\mid A\in {\mathfrak {P}}(S)\}}$ ${\displaystyle x^{*}}$ ${\displaystyle x^{*}}$ ${\displaystyle (\;)_{x^{*}}}$ ${\displaystyle A_{x^{*}}\in {\mathfrak {I}}_{x^{*}}}$ ${\displaystyle x^{*}}$ ${\displaystyle (A)_{x^{*}}}$ ${\displaystyle A\in {\mathfrak {P}}(S)}$ ${\displaystyle x^{*}}$ ${\displaystyle (a_{1},\cdots ,a_{n})_{x^{*}}:=(\{a_{1},\cdots ,a_{n}\})_{x^{*}}}$ ${\displaystyle a_{1},\cdots ,a_{n}\in S,n\in \mathbb {N} ,}$ ${\displaystyle x^{*}}$ ${\displaystyle (a)_{x^{*}}}$ ${\displaystyle a\in S}$ ${\displaystyle x^{*}}$

comment

• ${\displaystyle \emptyset }$is usually not an ideal, but because it is advantageous for ideal arithmetic, it should also be a spurious main ideal here , if . ${\displaystyle (\emptyset )_{x^{*}}=\bigcap {\mathfrak {I}}_{x^{*}}}$ ${\displaystyle x^{*}}$ ${\displaystyle (\emptyset )_{x^{*}}=\emptyset }$
• To distinguish between ideals and any subsets of , the ideals, in contrast to any subsets, are given a corresponding index in the following. ${\displaystyle S}$

Ideal associations

Up are two two-digit operations ${\displaystyle {\mathfrak {I}}_{x^{*}}}$

${\displaystyle \vee _{x^{*}}:{\mathfrak {I}}_{x^{*}}\times {\mathfrak {I}}_{x^{*}}\to {\mathfrak {I}}_{x^{*}},(A_{x^{*}},B_{x^{*}})\mapsto A_{x^{*}}\vee _{x^{*}}B_{x^{*}}:=(A_{x^{*}}\cup B_{x^{*}})_{x^{*}},}$

${\displaystyle \wedge _{x^{*}}:{\mathfrak {I}}_{x^{*}}\times {\mathfrak {I}}_{x^{*}}\to {\mathfrak {I}}_{x^{*}},(A_{x^{*}},B_{x^{*}})\mapsto A_{x^{*}}\wedge _{x^{*}}B_{x^{*}}:=(A_{x^{*}}\cap B_{x^{*}})_{x^{*}},}$

given so that it forms a complete association , the association of the ideals of . It is the -Idealverbindung , the -Idealdurchschnitt . ${\displaystyle ({\mathfrak {I}}_{x^{*}},\vee _{x^{*}},\wedge _{x^{*}})}$ ${\displaystyle x^{*}}$ ${\displaystyle (S,\cdot )}$ ${\displaystyle \vee _{x^{*}}}$ ${\displaystyle x^{*}}$ ${\displaystyle \wedge _{x^{*}}}$ ${\displaystyle x^{*}}$

As for all envelope systems, the following also applies to every ideal system: ${\displaystyle x^{*}}$

${\displaystyle \forall A,B\in {\mathfrak {P}}(S):}$

${\displaystyle A_{x^{*}}\wedge _{x^{*}}B_{x^{*}}=A_{x^{*}}\cap B_{x^{*}}.}$

Algebraic Idea Operators

${\displaystyle ({\mathfrak {I}}_{x^{*}},\vee _{x^{*}},\wedge _{x^{*}})}$is algebraic if and only if is algebraic , that is ${\displaystyle (\;)_{x^{*}}}$

${\displaystyle \forall a\in S\;\forall A\in {\mathfrak {P}}(S):}$

${\displaystyle A\neq \emptyset }$ and ${\displaystyle a\in (A)_{x^{*}}\implies \exists a_{1},\cdots ,a_{n}\in A;n\in \mathbb {N} :a\in (a_{1},\cdots ,a_{n})_{x^{*}}.}$

Denotes the power of the crowd , then exists with ${\displaystyle |A|}$ ${\displaystyle A}$

${\displaystyle (\;)_{x_{s}^{*}}:{\mathfrak {P}}(S)\to {\mathfrak {P}}(S),A\mapsto (A)_{x_{s}^{*}}:=\bigcup \{(N)_{x^{*}}\mid N\subseteq A,|N|\in \mathbb {N} _{0}\},}$

always an algebraic -idea operator to it . ${\displaystyle x^{*}}$ ${\displaystyle (\;)_{x^{*}}}$

x ideal operators

The ideal multiplication ${\displaystyle x^{*}}$

${\displaystyle \cdot _{x^{*}}:{\mathfrak {I}}_{x^{*}}\times {\mathfrak {I}}_{x^{*}}\to {\mathfrak {I}}_{x^{*}},(A_{x^{*}},B_{x^{*}})\mapsto A_{x^{*}}\cdot _{x^{*}}B_{x^{*}}:=(A_{x^{*}}\cdot B_{x^{*}})_{x^{*}},}$

has the property characteristic of ideals

${\displaystyle \forall A_{x^{*}},B_{x^{*}}\in {\mathfrak {I}}_{x^{*}}:}$

${\displaystyle B_{x^{*}}\cdot _{x^{*}}A_{x^{*}}\subseteq A_{x^{*}},}$

In general, however, it does not yet offer enough properties to be able to investigate properly. On the other hand, the following class of ideal operators has proven to be well suited for a general ideal theory . ${\displaystyle (S,\cdot )}$ ${\displaystyle x^{*}}$

definition

So-called -Idea operators or -operators are -Idea operators in which translations ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle (\;)_{x}}$ ${\displaystyle x^{*}}$

${\displaystyle \forall t\in S:}$

${\displaystyle \vartheta _{t}\colon S\to S,a\mapsto \vartheta _{t}(a):=a\cdot t,}$

As with topological closure operators, " continuous " are :

${\displaystyle \forall t\in S\;\forall A\in {\mathfrak {P}}(S):}$

${\displaystyle \vartheta _{t}{\bigl (}(A)_{x}{\bigr )}\subseteq {\bigl (}\vartheta _{t}(A){\bigr )}_{x}}$

with for each and everyone . ${\displaystyle \vartheta _{t}(A):=\{\vartheta _{t}(a)\mid a\in A\}=A\cdot \{t\}}$ ${\displaystyle t\in S}$ ${\displaystyle A\in {\mathfrak {P}}(S)}$

properties

• With every -idea operator there is also an -idea operator. ${\displaystyle x}$ ${\displaystyle (\;)_{x}}$ ${\displaystyle (\;)_{x_{s}}}$ ${\displaystyle x}$
• For every -ideal operator on even follows ${\displaystyle x}$ ${\displaystyle (\;)_{x}}$ ${\displaystyle (S,\cdot )}$

${\displaystyle \forall A,B\in {\mathfrak {P}}(S):}$

${\displaystyle (A)_{x}\cdot B\subseteq (A\cdot B)_{x}.}$

• The two-sided ideals of a semigroup are exactly the kernels of certain semigroup homomorphisms of , and it holds ${\displaystyle x}$ ${\displaystyle (S,\cdot )}$ ${\displaystyle (S,\cdot )}$

${\displaystyle \forall A,B\in {\mathfrak {P}}(S):}$

${\displaystyle (A)_{x}\cdot _{x}(B)_{x}=(A\cdot B)_{x}.}$

• A two-sided ideal system forms a (commutative,) associative, quasi-logical and complete multiplicative association . ${\displaystyle x}$ ${\displaystyle ({\mathfrak {I}}_{x},\vee _{x},\wedge _{x},\cdot _{x})}$
• Likewise, for two-sided ideals there is such a multiplicative lattice, which is also always algebraic. ${\displaystyle ({\mathfrak {I}}_{x_{s}},\vee _{x_{s}},\wedge _{x_{s}},\cdot _{x_{s}})}$ ${\displaystyle x}$

Remarks

• Any -ideal operator always induces an -ideal operator, so that -ideal operators are also of a very general nature. ${\displaystyle x^{*}}$ ${\displaystyle x}$ ${\displaystyle x}$
• Another, more abstract approach to a general ideal theory is the description of ideal systems using appropriate multiplicative associations.
• As a rule, terms from “classical” ideal theory, such as maximum ideal , prime ideal , etc., can be used for -ideals without any problems . ${\displaystyle x}$

r -Ideal operators

definition

An -Ideaoperator auf is an -Idealoperator, which is also translation closed , so ${\displaystyle r}$ ${\displaystyle (\;)_{r}}$ ${\displaystyle (S,\cdot )}$ ${\displaystyle x}$

${\displaystyle \forall t\in S\;\forall A_{r}\in {\mathfrak {I}}_{r}:}$

${\displaystyle \vartheta _{t}(A_{r})\in {\mathfrak {I}}_{r},}$

and for which also applies:

${\displaystyle \forall a\in S:}$

${\displaystyle (a)_{r}=\{a\}\cup \vartheta _{a}(S).}$

properties

• For every translation-completed -ideal operator on even follows ${\displaystyle x}$ ${\displaystyle (\;)_{x}}$ ${\displaystyle (S,\cdot )}$

${\displaystyle \forall t\in S\;\forall A\in {\mathfrak {P}}(S):}$

${\displaystyle \vartheta _{t}{\bigl (}(A)_{x}{\bigr )}={\bigl (}\vartheta _{t}(A){\bigr )}_{x}.}$

• If a unit has 1, then every translation-completed -idea operator is already an -idea operator and ${\displaystyle (S,\cdot )}$ ${\displaystyle x}$ ${\displaystyle (\;)_{x}}$ ${\displaystyle (S,\cdot )}$ ${\displaystyle r}$

${\displaystyle \forall a\in S:}$

${\displaystyle \left(1\right)_{x}=S}$ and ${\displaystyle (a)_{x}=\vartheta _{a}(S)=S\cdot \{a\}.}$

• ${\displaystyle (\;)_{r_{s}}}$is also an ideal operator. ${\displaystyle r}$
• Every two-sided main ideal is a multiplication ideal , that is ${\displaystyle r}$

${\displaystyle \forall a\in S\;\forall A_{r}\in {\mathfrak {I}}_{r}:}$

${\displaystyle A_{r}\subset (a)_{r}\iff \exists B_{r}\in {\mathfrak {I}}_{r}:B_{r}\cdot _{r}(a)_{r}=A_{r}\neq (a)_{r}.}$

• A two-sided one can be shortened , so ${\displaystyle (a)_{r}}$ ${\displaystyle ({\mathfrak {I}}_{r},\vee _{r},\wedge _{r},\cdot _{r})}$

${\displaystyle \forall a\in S\;\forall A_{r},B_{r}\in {\mathfrak {I}}_{r}:}$

${\displaystyle A_{r}\cdot _{r}(a)_{r}=B_{r}\cdot _{r}(a)_{r}\implies A_{r}=B_{r},}$

if in can be shortened . ${\displaystyle a\in S}$ ${\displaystyle (S,\cdot )}$

comment

• ${\displaystyle r}$Ideal systems have all the essential properties of the ideal systems of rings, which is why they allow a good investigation of the divisibility relationships in . ${\displaystyle d}$ ${\displaystyle (S,\cdot )}$

literature

• H. Examiner: Investigations into the divisibility properties of bodies . In: J. reine angew. Math. Band 168 , 1932, pp. 1-36 .
• K. E. Aubert: Theory of x-ideals . In: Acta Math. Band 107 , 1962, pp. 1-52 .
• I. Fleischer: Equivalence of x-systems and m-lattices . In: Colloquia Mathematica Societatis Janos Bolyai . 33. Contributions to Lattice Theory, Szeged, 1980. North Holland , Amsterdam / Oxford / New York 1983, pp. 381-400 .
• P. Lorenzen: Abstract foundation of the multiplicative ideal theory . In: Math. Z. Band 45 , 1939, pp. 533-553 .
• M. Ward, R. P. Dilworth: The lattice theory of ova . In: Ann. Math. Band 40 , 1939, pp. 600-608 .
• L. Fuchs :: Partly ordered algebraic structures . Vandenhoeck & Ruprecht , Göttingen 1966.
• G. Birkhoff: Lattice Theory . 3. Edition. American Mathematical Society , Providence (R. I.) 1973.