On Algebraic Multi-Ring Spaces

arXiv:math/0510478v1 [math.GM] 22 Oct 2005

On Algebraic Multi-Ring Spaces Linfan Mao (Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080)

Abstract: A Smarandache multi-space is a union of n spaces A1 , A2 , · · · , An with some additional conditions holding. Combining Smarandache multispaces with rings in classical ring theory, the conception of multi-ring spaces is introduced. Some characteristics of a multi-ring space are obtained in this paper

Key words: ring, multi-space, multi-ring space, ideal subspace chain.

Classification: AMS(2000) 16U80, 16W25 1. Introduction The notion of multi-spaces is introduced by Smarandache in [6] under his idea of hybrid mathematics: combining different fields into a unifying field([7]), which is defined as follows.

Definition 1.1 For any integer i, 1 ≤ i ≤ n let Ai be a set with ensemble of law Li , and the intersection of k sets Ai1 , Ai2 , · · · , Aik of them constrains the law I(Ai1 , Ai2 , · · · , Aik ). Then the union of Ai , 1 ≤ i ≤ n Ae

n [

=

Ai

i=1

is called a multi-space.

As we known, a set R with two binary operation + and ◦, denoted by (R ; +, ◦), is said to be a ring if for ∀x, y ∈ R, x + y ∈ R, x ◦ y ∈ R, the following conditions hold. (i) (R ; +) is an abelian group; (ii) (R ; ◦) is a semigroup; (iii) For ∀x, y, z ∈ R, x ◦ (y + z) = x ◦ y + x ◦ z and (x + y) ◦ z = x ◦ z + y ◦ z.

By combining Smarandache multi-spaces with rings, a new kind of algebraic structure called multi-ring space is found, which is defined in the following. m e = S R be a complete multi-space with double binary operaDefinition 1.2 Let R i i=1

e O(R)

tion set = {(+i , ×i ), 1 ≤ i ≤ m}. If for any integers i, j, i 6= j, 1 ≤ i, j ≤ m, e (Ri ; +i , ×i ) is a ring and for ∀x, y, z ∈ R, 1

(x ×i y) ×j z = x ×i (y ×j z)

(x +i y) +j z = x +i (y +j z), and x ×i (y +j z) = x ×i y +j x ×i z,

(y +j z) ×i x = y ×i x +j z ×i x

e is called a multi-ring space. If for any if all their operation results exist, then R e is called a multi-filed space. integer 1 ≤ i ≤ m, (R; +i , ×i ) is a filed, then R m e ⊂ R e and O(S) e ⊂ O(R), e if S e is also a e = S R , let S For a multi-ring space R i i=1

e , then call S e a multi-ring multi-ring space with double binary operation set O(S) e We have the following criterions for the multi-ring subspaces. subspace of R. The subject of this paper is to find some characteristics of a multi-ring space. For terminology and notation not defined here can be seen in [1], [5], [12] for algebraic terminologies and in [2], [6] − [11] for multi-spaces and logics.

2. Characteristics of a multi-ring space First, we have the following result for multi-ring subspace of a multi-ring space. m e ⊂ R e with O(S) e ⊂ e = S R , a subset S Theorem 2.1 For a multi-ring space R i i=1

e O(R) eT

e R

is a multi-ring subspace of if and only if for any integer k, 1 ≤ k ≤ m, T Rk ; +k , ×k ) is a subring of (Rk ; +k , ×k ) or Se Rk = ∅.

(S

Proof For any integer k, 1 ≤ k ≤ m, if (Se

eT

or S

Rk = ∅, then since Se =

m T S (Se R

i=1

i ),

T

Rk ; +k , ×k ) is a subring of (Rk ; +k , ×k )

we know that Se is a multi-ring subspace

by definition of a multi-ring space. s S e with double binary operation Sij is a multi-ring subspace of R Now if Se = j=1

e O(S)

set = {(+ij , ×ij ), 1 ≤ j ≤ s}, then (Sij ; +ij , ×ij ) is a subring of (Rij ; +ij , ×ij ).

T e Therefore, for any integer j, 1 ≤ j ≤ s, Sij = Rij S. But for other integer l ∈ T e {i; 1 ≤ i ≤ m} \ {ij ; 1 ≤ j ≤ s}, S Sl = ∅. ♮ Applying the criterions for subrings of a ring, we get the following result. m e ⊂ R e with O(S) e ⊂ e = S R , a subset S Theorem 2.2 For a multi-ring space R i i=1

e is a multi-ring subspace of R e if and only if for any double binary operations O(R) T e (S e R ; + ) ≺ (R ; + ) and (S; e × ) is complete. (+j , ×j ) ∈ O(S), j j j j j

Proof According to Theorem 2.1, we know that Se is a multi-ring subspace if and T only if for any integer i, 1 ≤ i ≤ m, (Se Ri ; +i, ×i ) is a subring of (Ri ; +i , ×i ) or T Se Ri = ∅. By a well known criterions for subrings of a ring (see also [5]), we know 2

T

that (Se Ri ; +i , ×i ) is a subring of (Ri ; +i , ×i ) if and only if for any double binary e (S e T R ; + ) ≺ (R ; + ) and (S; e × ) is a complete set. operations (+j , ×j ) ∈ O(S), j j j j j This completes the proof. ♮ We use the ideal subspace chain of a multi-ring space to characteristic its strucm e = S R with double ture properties. An ideal subspace Ie of a multi-ring space R i i=1

binary operation set conditions:

e O(R)

is a multi-ring subspace of

e R

satisfying the following

e (i) Ie is a multi-group subspace with operation set {+| (+, ×) ∈ O(I)}; e a ∈ Ie and (+, ×) ∈ O(I), e r × a ∈ Ie and a × r ∈ Ie if their (ii) for any r ∈ R, operation results exist. m e O(I) e ⊂ O(R) e of a multi-ring space R e = S R Theorem 2.3 A subset Ie with O(I), i i=1

e = {(+ , × )| 1 ≤ i ≤ m} is an ideal subspace with double binary operation set O(R) i Ti e if and only if for any integer i, 1 ≤ i ≤ m, (I Ri , +i , ×i ) is an ideal of the ring T (Ri , +i , ×i ) or Ie Ri = ∅.

Proof By definition of an ideal subspace, the necessity of the condition is obvious. e e with binary operFor the sufficiency, denote by R(+, ×) the set of elements in R T T e ations + and ×. If there exists an integer i such that I Ri 6= ∅ and (Ie Ri , +i , ×i ) T is an ideal of (Ri , +i , ×i ), then for ∀a ∈ Ie Ri , ∀ri ∈ Ri , we know that ri ×i a ∈ Ie

\

a ×i ri ∈ Ie

Ri ;

\

Ri .

e e Notice that R(+ i , ×i ) = Ri . Therefore, we get that for ∀r ∈ R,

r ×i a ∈ Ie

\

Ri ; and a ×i r ∈ Ie

\

Ri ,

e if their operation result exist. Whence, Ie is an ideal subspace of R. ♮ e e An ideal subspace I of a multi-ring space R is said maximal if for any ideal e ⊇ Ie′ ⊇ I, e then Ie′ = R e or Ie′ = I. e For any order of the double subspace Ie′ , if R m e of a multi-ring space R e = S R , not loss of generality, binary operation set O(R) i i=1

assume it being (+1 , ×1 ) ≻ (+2 , ×2 ) ≻ · · · ≻ (+m , ×m ), we can define an ideal e by the following programming. subspace chain of R (i) Construct the ideal subspace chain e ⊃R e ⊃R e ⊃ ··· ⊃ R e R 11 12 1s1

e is a maximal ideal subspace under the double binary operation (+1 , ×1 ), where R 11 e and in general, for any integer i, 1 ≤ i ≤ m − 1, R e of R 1(i+1) is a maximal ideal e subspace of R1i . (ii) If the ideal subspace

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e ⊃R e ⊃R e ⊃ ··· ⊃ R e e e R 11 12 1s1 ⊃ · · · ⊃ Ri1 ⊃ · · · ⊃ Risi

has been constructed for (+1 , ×1 ) ≻ (+2 , ×2 ) ≻ · · · ≻ (+i , ×i ), 1 ≤ i ≤ m − 1, then e construct an ideal subspace chain of R isi e e e e ⊃R R isi (i+1)1 ⊃ R(i+1)2 ⊃ · · · ⊃ R(i+1)s1 e e under the operations (+i+1 , ×i+1 ), where R (i+1)1 is a maximal ideal subspace of Risi e e and in general, R (i+1)(i+1) is a maximal ideal subspace of R(i+1)j for any integer e under (+ , × ) ≻ (+ , × ) ≻ j, 1 ≤ j ≤ si − 1. Define the ideal subspace chain of R 1 1 2 2 · · · ≻ (+i+1 , ×i+1 ) being e ⊃R e ⊃ ··· ⊃ R e e e e e R 11 1s1 ⊃ · · · ⊃ Ri1 ⊃ · · · ⊃ Risi ⊃ R(i+1)1 ⊃ · · · ⊃ R(i+1)si+1 .

Similar to a multi-group space([3]), we have the following results for the ideal subspace chain of a multi-ring space. m e = S R , its ideal subspace chain only has Theorem 2.4 For a multi-ring space R i i=1

finite terms if and only if for any integer i, 1 ≤ i ≤ m, the ideal chain of the ring (Ri ; +i , ×i ) has finite terms, i.e., each ring (Ri ; +i , ×i ) is an Artin ring. − → e Proof Let the order of double operations in O (R) be (+1 , ×1 ) ≻ (+2 , ×2 ) ≻ · · · ≻ (+m , ×m ) and a maximal ideal chain in the ring (R1 ; +1 , ×1 ) is R1 ≻ R11 ≻ · · · ≻ R1t1 .

Calculate e R

11

=

e \ {R R

1

\ R11 } = R11

m [ [

(

)Ri ,

i=2

e =R e \ {R \ R } = R R 12 11 11 12 12

m [ [

(

)Ri ,

i=2

·················· e R

1t1

=

e R

1t1

\ {R1(t1 −1) \ R1t1 } = R1t1

m [ [

(

)Ri .

i=2

According to Theorem 3.10, we know that e ⊃R e ⊃R e ⊃ ··· ⊃ R e R 11 12 1t1

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e under the double binary operation (+ , × ). is a maximal ideal subspace chain of R 1 1 In general, for any integer i, 1 ≤ i ≤ m − 1, assume

Ri ≻ Ri1 ≻ · · · ≻ Riti is a maximal ideal chain in the ring (R(i−1)ti−1 ; +i , ×i ). Calculate e =R R ik ik

m [

[

(

e )R ik

\

Ri

j=i+1

Then we know that e e e e R (i−1)ti−1 ⊃ Ri1 ⊃ Ri2 ⊃ · · · ⊃ Riti e is a maximal ideal subspace chain of R (i−1)ti−1 under the double operation (+i , ×i )

by Theorem 3.10. Whence, if for any integer i, 1 ≤ i ≤ m, the ideal chain of the ring (Ri ; +i , ×i ) has finite terms, then the ideal subspace chain of the multi-ring space e only has finite terms and if there exists one integer i such that the ideal chain of R 0 the ring (Ri0 , +i0 , ×i0 ) has infinite terms, then there must be infinite terms in the e ideal subspace chain of the multi-ring space R. ♮.

A multi-ring space is called an Artin multi-ring space if each ideal subspace chain only has finite terms. We have the following corollary by Theorem 3.11. m e = e = S with double binary operation set O(R) Corollary 2.1 A multi-ring space R i=1

{(+i , ×i )| 1 ≤ i ≤ m} is an Artin multi-ring space if and only if for any integer i, 1 ≤ i ≤ m, the ring (Ri ; +i , ×i ) is an Artin ring. m e = e = S with double binary operation set O(R) For a multi-ring space R i=1

{(+i , ×i )| 1 ≤ i ≤ m}, an element e is an idempotent element if e2× = e × e = e for a e We define the directed sum Ie of two ideal double binary operation (+, ×) ∈ O(R). e e subspaces I1 and I2 as follows: S

(i) Ie = Ie1 Ie2 ; T T (ii) Ie1 Ie2 = {0+ }, or Ie1 Ie2 = ∅, where 0+ denotes an unit element under the operation +.

Denote the directed sum of Ie1 and Ie2 by L

Ie = Ie1

M

Ie2 .

e then Ie is called If for any Ie1 , Ie2 , Ie = Ie1 Ie2 implies that Ie1 = Ie or Ie2 = I, non-reducible. We have the following result for the Artin multi-ring space similar to a well-known result for the Artin ring (see [12]).

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m e = S R with double binary operation Theorem 2.5 Any Artin multi-ring space R i i=1

e = {(+ , × )| 1 ≤ i ≤ m} is a directed sum of finite non-reducible ideal set O(R) i i subspaces, and if for any integer i, 1 ≤ i ≤ m, (Ri ; +i, ×i ) has unit 1×i , then e = R

si m M M

(Ri ×i eij )

(

[

(eij ×i Ri )),

i=1 j=1

where eij , 1 ≤ j ≤ si are orthogonal idempotent elements of the ring Ri . f the set of ideal subspaces which can not be represented by Proof Denote by M e According to Theorem 3.11, there is a directed sum of finite ideal subspaces in R. f . It is obvious that Ie is reducible. a minimal ideal subspace Ie0 in M 0 e e e f and Ie 6∈ M f . Therefore, Ie and Ie can Assume that I0 = I1 + I2 . Then Ie1 6∈ M 2 1 2 be represented by directed sums of finite ideal subspaces. Whence, Ie0 can be also f. represented by a directed sum of finite ideal subspaces. Contradicts that Ie0 ∈ M Now let e = R

s M i=1

Iei ,

where each Iei , 1 ≤ i ≤ s, is non-reducible. Notice that for a double operation (+, ×), e has the form each non-reducible ideal subspace of R (e × R(×))

[

(R(×) × e), e ∈ R(×).

e such that Whence, we know that there is a set T ⊂ R e = R

M

(e × R(×))

[

(R(×) × e).

e e∈T, ×∈O(R)

e and the unit 1 , assume that For any operation × ∈ O(R) ×

1× = e1 ⊕ e2 ⊕ · · · ⊕ el , ei ∈ T, 1 ≤ i ≤ s.

Then ei × 1× = (ei × e1 ) ⊕ (ei × e2 ) ⊕ · · · ⊕ (ei × el ).

Therefore, we get that ei = ei × ei = e2i and ei × ej = 0i for i 6= j. e That is, ei , 1 ≤ i ≤ l, are orthogonal idempotent elements of R(×).

Notice that e R(×) = Rh for some integer h. We know that ei , 1 ≤ i ≤ l are orthogonal idempotent elements of the ring (Rh , +h , ×h ). Denoted by ehj for ej , 1 ≤ j ≤ l. Consider all e we get that units in R,

6

e R

This completes the proof.

=

si m M M

(Ri ×i eij )

(

[

(eij ×i Ri )).

i=1 j=1

♮

Corollary 2.2([12]) Any Artin ring (R ; +, ×) is a directed sum of finite ideals, and if (R ; +, ×) has unit 1× , then R=

s M

Ri ei ,

i=1

where ei , 1 ≤ i ≤ s are orthogonal idempotent elements of the ring (R; +, ×).

3. Open problems for a multi-ring space Similar to Artin multi-ring space, we can also define Noether multi-ring spaces, simple multi-ring spaces, half-simple multi-ring spaces, · · ·, etc.. The open problems for these new algebraic structure are as follows.

Problem 3.1 Call a ring R a Noether ring if its every ideal chain only has finite e if its every ideal multi-ring subspace chain terms. Similarly, for a multi-ring space R, only has finite terms, it is called a Noether multi-ring space. Whether can we find its structures similar to Corollary 2.2 and Theorem 2.5?

Problem 3.2 Similar to ring theory, define a Jacobson or Brown-McCoy radical for multi-ring spaces and determine their contribution to multi-ring spaces.

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[6] F.Smarandache, Mixed noneuclidean geometries, eprint arXiv: math/0010119, 10/2000.

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[8] F.Smarandache, Neutrosophy, a new Branch of Philosophy, Multi-Valued Logic, Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 297384.

[9] F.Smarandache, A Unifying Field in Logic: Neutrosophic Field, Multi-Valued Logic, Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 385-438.

[10]] W.B.Vasantha Kandasamy, Bialgebraic structures and Smarandache bialgebraic structures, American Research Press, 2003.

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