Rigid and Non-Rigid Mathematical Theories: the Ring Z Is Nearly Rigid
aa r X i v : . [ m a t h . G M ] M a y Rigid and Non-Rigid MathematicalTheories : the Ring Z Is Nearly Rigid
Elem´er E Rosinger
Department of Mathematicsand Applied MathematicsUniversity of PretoriaPretoria0002 South [email protected]
Dedicated to Marie-Louise Nykamp
Abstract
Mathematical theories are classified in two distinct classes : rigid , andon the other hand, non-rigid ones. Rigid theories, like group the-ory, topology, category theory, etc., have a basic concept - given forinstance by a set of axioms - from which all the other concepts aredefined in a unique way. Non-rigid theories, like ring theory, certaingeneral enough pseudo-topologies, etc., have a number of their con-cepts defined in a more free or relatively independent manner of oneanother, namely, with compatibility conditions between them only. Asan example, it is shown that the usual ring structure on the integers Z is not rigid, however, it is nearly rigid.
0. Introduction
Rigid theories, like group theory, topology, category theory, etc., havea basic concept - given for instance by a set of axioms - from whichall the other concepts are defined in a unique way. Non-rigid theories,like ring theory, certain general enough pseudo-topologies, etc., havea number of their concepts defined in a more free or relatively inde-pendent manner of one another, namely, with compatibility conditionsbetween them only. 1ne can note that even in Algebra there are nonrigid mathematicalstructures. For instance, let ( R, + , . ) be a ring. Then in principle, nei-ther the addition ”+” determines the multiplication ”.”, nor multipli-cation determines addition. Instead, they are relatively independentof one another, and only satisfy the usual compatibility conditions,namely, the distributivity of multiplication with respect to addition.On the contrary, in groups ( G, ⋄ ), all concepts are defined uniquelybased eventually on the underlying set G and the binary operation ⋄ no longer constitute a usual Eilenberg- Mac Lane category, but one which is more general, [8,12].As it happens, the rigid structure of the usual Hausdorff-Kuratowski-Bourbaki, or in short, HKB concept of topology is also one of the rea-sons for a number of its important deficiencies, such as for instance,that the category of such topological spaces is not Cartesian closed.Spaces (Ω , M , µ ) with measure, where Ω is the underlying set, M is a σ -algebra on it, and µ : M −→ R is a σ -additive measure, are furtherexamples of non-rigid structures, since for a given (Ω , M ), there canin general be infinitely many associated µ .Topological groups, or even topological vector spaces, are typically2on-rigid structures. Indeed, on an arbitrary group, or even vectorspace, there may in general be many compatible topologies, and evenHausdorff topologies.Obviously, an important advantage of a rigid mathematical structure,and in particular, of the usual HKB concept of topology, is a simplic-ity of the respective theoretical development. Such simplicity comesfrom the fact that one can start with only one single concept, like forinstance the open sets in the case of HKB topologies, and then basedon that concept, all the other concepts can be defined in a uniquemanner.Consequently, the impression may be created that one has managedto develop a universal theory in the respective discipline, universal inthe sense that there may not be any need for alternative theories inthat discipline, as for instance is often the perception about the HKBtopology.The disadvantage of a rigid mathematical structure is in a consequentbuilt in lack of flexibility regarding the interdependence of the vari-ous concepts involved, since each of them, except for a single startingconcept, are determined uniquely in terms of that latter one. And inthe case of the HKB topologies this is manifested, among others, inthe difficulties related to dealing with suitable topologies on spacesof continuous functions, that is, in the failure of the category of suchtopological spaces to be Cartesian closed.Non-rigid mathematical structures, and in particular, certain generalenough pseudo-topologies, can manifest fewer difficulties coming froma lack of flexibility.A disadvantage of such non-rigid mathematical structures - as for in-stance with various approaches to pseudo-topologies - is in the largevariety of ways the respective theories can be set up. Also, their re-spective theoretical development may turn out to be more complexthan is the case with rigid mathematical structures.Such facts can lead to the impression that one could not expect to finda universal enough non-rigid mathematical structure in some givendiscipline, and for instance, certainly not in the realms of pseudo-3opologies.As it happens so far in the literature on pseudo-topologies, there seemsnot to be a wider and explicit enough awareness about the followingtwo facts • one should rather use non-rigid structures in order to avoid thedifficulties coming from the lack of flexibility of the rigid conceptof usual HKB topology, • the likely consequence of using non-rigid structures is the lackof a sufficiently universal concept of pseudo-topology.As it happens, such a lack of awareness leads to a tendency to developmore and more general concepts of pseudo-topology, hoping to reacha sufficiently universal one, thus being able to replace once and for allthe usual HKB topology with ”THE” one and only ”winning” conceptof pseudo-topology.Such an unchecked search for increased generality, however, may eas-ily lead to rather meagre theories.It also happens in the literature that, even if mainly intuitively, whensetting up various concepts of pseudo-topology a certain restraint ismanifested when going away from a rigid theory towards some non-rigid ones. And certainly, the reason for such a restraint is that onewould like to hold to the advantage of rigid theories which are moresimple to develop than the non-rigid ones.Amusingly, the precedent in Geometry, happened two centuries ear-lier, is missed from the view both by those who hold to the usualHKB concept of topology, as well as by those trying to set up a gen-eral enough pseudo-topology which hopefully may be universal. Afterall, having by now gotten accustomed that, in fact so fortunately, Ge-ometry can mean many things in different situations, it may perhapsbe appropriate to accept a similar view regarding Topology ...Lastly, let us note that in modern Mathematics it is ”axiomatic” thattheories are built as axiomatic systems .4s it happens, however, ever since the early 1930s and G¨odel’s In-completeness Theorems, we cannot disregard the consequent deeplyinherent limitation of all axiomatic mathematical theories.And that limitation cannot be kept away from nonrigid mathemati-cal structures either, since such structures are also built as axiomatictheories.And that G¨odelian limitation comes to further suggest the answer tothe issue of what is Topology, is it the HKB one, or is it one or anotherpseudo-topology ?And the answer is simple indeed : the rigid HKB concept of topologymay be just as little unique, as that of Geometry proved to be twocenturies earlier ...Regarding Mathematics in general, fortunately, two possible furtherdevelopments, away from that G¨odelian limitation of all axiomatic the-ories, have recently appeared, even if they are not yet clearly enoughin the general mathematical awareness. Namely, self-referential ax-iomatic mathematical theories, and perhaps even more surprisingly, inconsistent axiomatic mathematical theories, [14].
1. Rings Are Non-Rigid Structures
In Group Theory, given any group ( G, ⋄ ), be it commutative or not, allthe concepts of the theory will in the last analysis be uniquely definedby the set G and the binary operation ⋄ .On the other hand, in Ring Theory, given a ring ( R, + , . ), the binaryoperation ′′ + ′′ of addition, does not in general determine uniquely thebinary operation ′′ . ′′ of multiplication. Instead, they are only supposedto satisfy certain compatibility relations, namely, the distributivity ofmultiplication with respect to addition.And rather simple examples show that, given a commutative group,there can be more than one ring multiplication on it.Indeed, let M n ( R ), with n ≥
2, be the set of n × n matrices of real5umbers, and consider on it the commutative group structure givenby the usual addition ′′ + ′′ of matrices.The following two different ring structures can be defined on M n ( R ).First, let ( M n ( R ) , + , . ), where ′′ . ′′ is the usual noncommutative mul-tiplication of square matrices. Second, let ( M n ( R ) , + , ∗ ) where ′′ ∗ ′′ isthe term by term multiplication of matrices, namely, given the matri-ces A = ( a i, j | ≤ i, j ≤ n ) , B = ( b i, j | ≤ i, j ≤ n )we have A.B = C , where C = ( c i, j | ≤ i, j ≤ n )with c i, j = a i, j .b i, j And these two ring structures are indeed different, although their un-derlying commutative group structure is the same. For instance, thefirst one is noncommutative, while the second one is commutative.Furthermore, the first one has as unit element the square matrix withthe diagonal 1, and with all the other elements 0, while the unit ele-ment in the second one is the matrix with all the elements 1.Consequently, Ring Theory is indeed non-rigid .
2. The Ring Z Is Nearly Rigid
Let ′′ + ′′ denote the usual addition on Z while ′′ . ′′ denotes the usualmultiplication on it. Further, for a given integer(2.1) a ∈ Z let us consider on Z the binary operation ′′ ∗ ′′ defined by62.2) n ∗ m = a.n.m, n, m ∈ Z Lemma 2.1. ( Z , + , ∗ ) is a commutative ring . Proof.
We have for n, m, k ∈ Z the relations n ∗ ( m ∗ k ) = a.n. ( m ∗ k ) = a.n. ( a.m.k ) = a.a.n.m.k while( n ∗ m ) ∗ k = a. ( n ∗ m ) .k = a. ( a.n.m ) .k = a.a.n.m.k thus the associativity of ′′ ∗ ′′ . Also we have the relations n ∗ ( m + k ) = a.n. ( m + k ) = ( a.n.m ) + ( a.n.k ) = ( n ∗ m ) + ( n ∗ k )hence the distributivity of ′′ ∗ ′′ with respect to ′′ + ′′ . (cid:3) Obviously, if a = 1, then ( Z , + , ∗ ) is the usual ring ( Z , + , . ). On theother hand, if a = −
1, then ( Z , + , ∗ ) has the somewhat surprisingmultiplication rule(2.3) n ∗ m = − n.m, n, m ∈ Z and we shall call this the alternate ring of the usual ring ( Z , + , . ). Lemma 2.2. ( Z , + , ∗ ) is a unital ring, if and only if a = ±
1, in which case it reducesto the usual ring ( Z , + , . ), or to its alternate, see (2.3).7 roof. Let u ∈ Z , such that u ∗ n = n, n ∈ Z then a.u.n = n, n ∈ Z thus in particular a.u = 1which, in view of (2.1), means that a = u = 1, or a = u = − (cid:3) Recalling now (2.1), (2.2), we obtain
Theorem 2.1.
All the commutative ring structures on the commutative group ( Z , +)are of the form ( Z , + , ∗ ), for suitable a ∈ Z . Proof.
Let be given any commutative ring ( Z , + , ◦ ), then we denote(2.4) a = 1 ◦ n, m ∈ Z , n, m >
0, then(2.5) n ◦ m = a.n.m Indeed 8 ◦ m = 1 ◦ (1 + . . . + 1) = a + . . . + a = a.mn ◦ m = (1 + . . . + 1) ◦ m = 1 ◦ m + . . . + 1 ◦ m = a.m + . . . + a.m = a.n.m Remark 2.1.
1. As noted in section 1, Ring Theory is non-rigid, since the commu-tative group structure of a ring does not in general determine uniquelythe ring multiplication.However, as seen in Theorem 2.1. above, the usual commutative groupstructure on Z does determine the commutative multiplication on it,except for a constant factor in (2.1), (2.2).As also seen in Lemma 2.1. above, the usual commutative group struc-ture on Z does further determine the commutative multiplication onit in case this multiplication has a unit element, except for the possi-bility of the alternate ring structure.2. The fact that on such a small set like Z , small in the sense of havingthe smallest infinite cardinal, there are not many significantly differ-ent ring structures on its usual commutative group need not come asa surprise. Indeed, in [9,11] it was shown that on N there are fewassociative binary operations which satisfy some rather natural andmild conditions.On the other hand, if we consider the commutative group ( M n ( Q ) , +)of n × n matrices of rational numbers with the usual addition of ma-trices, then as seen in section 1, two different ring structures can beassociated with that group. Yet the set M n ( Q ) has also the smallestinfinite cardinal. It may therefore be the case that in the mentionedresult in [9,11] the usual linear order on Z , and thus induced on N aswell, an order missing on M n ( Q ), plays a role.3. Obviously, the construction in (2.1), (2.2) can be applied to anarbitrary ring, and the result in Lemma 2.1. will still hold.In particular, the alternate in (2.3) can be defined far arbitrary rings.9s for the result in Lemma 2.2., its proof uses the fact that in a unitalring ( R, + , . ), the equation(2.6) a.u = 1 , a, u ∈ R should only have the solutions(2.7) a = u = ± Lemma 2.3.
Given a unital ring ( R, + , . ) with property (2.6), (2.7). If a ∈ R , thenthe ring ( R, + , ∗ ) obtained through (2.1), (2.2) is unital, if and only if a = ±