Chapter 1 What is a Ring? 1.1 Some Examples Considerthefollowingsets: 1. Z = {...,−1,0,1,2,...}–thesetofintegers. 2. Q = {a |a,b ∈ Z, b 6= 0}–thesetofrationalnumbers. b 3. M (R)–thesetof2×2matriceswithrealnumbersasentries. 2 4. 2Z = {...,−2,0,2,4,...}–thesetofevenintegers. 5. C(R)–thesetofcontinuousfunctionsfromRtoR. 6. Q[x] = {a xn + a xn−1 + ··· + a x + a |a ,...,a ∈ Q} – the set of n n−1 1 0 n 0 polynomialswithrationalcoefficients. 7. Z/6Z = {0¯,1¯,2¯,3¯,4¯,5¯}–thesetofcongruenceclassesinZmodulo6. Remember the very general definition of an algebraic structure as a set equipped withabinaryoperation,thatisaschemeforcombininganypairofelementsofthe set to produce a new element of the same set. All of the sets in our list above have binaryoperationsdefinedontheminnaturalandprobablymostlyfamiliarways. Ofcourseitispossibleforasettohavemorethanone“natural”binaryoperation definedonit. Algebra,initsbroadestsense,isthestudyofalgebraicstructures. Whatdoallthesixsetsdescribedabovehaveincommonasalgebraicstructures? Each of them is equipped with two binary operations called addition and mul- Z Q Z tiplication. In , and 2 we have the usual addition and multiplication of integers and rational numbers. In M (R) we have matrix addition and matrix 2 multiplication. InC(R)wehaveadditionandmultiplicationdefinedby ( f+g )(x) = f(x)+g(x), forallx ∈ Randallf,g ∈ C(R) |{z} | {z } +inC(R) +inR 1 ( f×g )(x) = f(x)×g(x), forallx ∈ Randallf,g ∈ C(R). |{z} | {z } ×inC(R) ×inR InQ[x]wehavetheusualadditionandmultiplicationofpolynomials,e.g. (x2 +2x+4)+(x3 −3x+2) = x3 +x2 −x+6, (x2 −2x+1)(x+5) = x3 +5x2 −2x2 −10x+x+5 = x3 +3x2 −9x+5. In Z/6Z the addition and multiplication are defined modulo 6, e.g. 4¯ + 5¯ = 3¯; 4¯ ×5¯ = 2¯,etc. Note: In each case the set under consideration is closed under the relevant opera- tionsofadditionandmultiplication;thismeansthatineachcasetheproductand sum of a pair of elements in a particular set also belong to that set. For example the set of odd integers is not closed under addition, since the sum of two odd integersisnotodd. ADDITION IN OUR EXAMPLES • All the above examples contain an identity element for addition, which we refer to as the zero element and write as 0. This element has the property that adding it to another element has no effect. The zero elements in our examplesare 1. Theinteger0 2. Therationalnumber0 (cid:18) (cid:19) 0 0 3. Thezeromatrix 0 0 4. Theinteger0 5. Thefunctionf : R −→ Rdefinedbyf(x) = 0, ∀x ∈ R 0 6. Thezeropolynomial0 7. Thecongruenceclass0¯ modulo6 • In each of our sets, every element has an additive inverse or “negative”. Two elements are additive inverses each other if their sum is the zero ele- ment. The fact that every element of a set has an additive inverse means thatsubtractioncanbedefinedintheset. • In all of our sets, addition is commutative, i.e. a+b = b+a for all pairs a andbofelements. MULTIPLICATION IN OUR EXAMPLES • The multiplication is commutative in all these examples except for M (R). 2 For2×2matricesAandB,theproductsABandBAneednotbeequal. 2 • Except for 2Z each of these examples contains an identity element for mul- tiplication, i.e. an element e for which e×a = a×e = a for all elements a oftheset;multiplyingbyehasnoeffect. Themultiplicativeidentitiesare 1. Theinteger1 2. Therationalnumber1 (cid:18) (cid:19) 1 0 3. Thematrix 0 1 4. Noidentityelementformultiplication 5. Thefunctionf : R −→ Rdefinedbyf(x) = 1forallx ∈ R 1 6. Thepolynomial1 7. Thecongruenceclass1¯ modulo6 • Two elements are multiplicative inverses of each other if their product is Q themultiplicativeidentityelement. In ,everyelementexcept0hasamul- tiplicative inverse, namely its reciprocal. All the other examples contain non-zeroelementswithoutmultiplicativeinverses. Thesevenalgebraicstructuresmentionedinthissectionareallexamplesofrings. 3 1.2 The Axioms of a Ring NOTE: In this section and throughout these lecture notes, please do not confuse the symbol R, which is used for a general ring, with the symbol R which is used forthesetofrealnumbers. Definition1.2.1 Aringisanon-emptysetRequippedwithtwobinaryoperationscalled addition(+)andmultiplication(×),satisfyingthefollowingproperties: Thefirstfourareconcernedwiththeoperationthatiscalledaddition. A1 Additionisassociative. (r+s)+t = r+(s+t)forallr,s,t ∈ R. A2 Additioniscommutative. r+s = s+rforallr,s ∈ R. A3 R contains an identity element for addition, denoted by 0 and called the R zeroelementofR. r+0 = 0 +r = rforallr ∈ R. R R A4 Every element of R has an inverse with respect to addition. (The additive inverseofrisoftendenotedby−r). Foreveryr ∈ R,thereexistsanelement−r ∈ Rforwhichr+(−r) = 0 . R NOTE : Axioms A1 to A4 could be summarized by saying that R is an abelian groupunderaddition. (Ifthisremarkisnothelpfulforyou,disregarditfornow). Themultiplicationoperationisrequiredonlytosatisfyonespecialcondition: M1 Multiplicationisassociative. (r×s)×t = r×(s×t)forallr,s,t ∈ R. The last two axioms are concerned with the manner in which the two operations mustinteract. D1 r×(s+t) = (r×s)+(r×t)forallr,s,t ∈ R. D2 (r+s)×t = (r×t)+(s×t)forallr,s,t ∈ R. -Distributivelawsformultiplicationoveraddition. REMARKS 1. Aringiscalledcommutativeifitsmultiplicationiscommutative. 4 2. A ring R is called unital or referred to as a ring with identity if it contains an identity element for multiplication. In this case we will denote the multi- plicative identity by 1 or just 1. We have already met one example of a R Z ringwithoutidentity,namelythering2 ofevenintegers. 3. The term “ring” was introduced by David Hilbert in the late 19th century, whenhereferredtoa“Zahlring”or“numberring”. Ourfirsttheoremaboutringsisthefollowingconsequenceoftheringaxioms. Theorem1.2.2 LetRbearing. ThenforallelementsrofRwehave 0 ×r = 0 andr×0 = 0 . R R R R i.e. multiplying any element of R by the zero element results in the zero element as the product. Proof: Letr ∈ R. Wehave (0 ×r)+(0 ×r) = (0 +0 )×r R R R R = 0 ×r. R Adding the additive inverse of the element 0 × r to both sides of this equation R gives 0 ×r = 0 . R R Asimilarargumentshowsthatr×0 = 0 . (cid:3) R R THREE REMARKS 1. The problem of deducing the truth of a statement like Theorem 1.2.2 from the axioms of a ring might be somewhat daunting. The proof may not be toohardtofollow,butcouldyouhavecomeupwithityourself? Ifyouwere tryingto,andyoudidn’tknowwheretostart,therearecertainobservations you could make that might help. There are seven axioms for rings - which might be likely to be helpful in proving the two (left and right) statements ofTheorem1.2.2? Well,thestatementisaboutmultiplicationandaboutthe zero element. According to the ring axioms, what is special about the zero element has to do with addition not multiplication. So it might seem likely that the statement in the theorem is essentially connected to the interaction of the addition and multiplication - the two axioms that deal with that are thedistributivelaws,somaybeweshouldnotbesosurprisedthatthesehave acrucialroleintheproof. 5 2. The next two remarks are about the philosophy of abstract algebra and the mechanisms by which the subject progresses. The definition of a ring con- sists of a list of technical properties, but the motivation for this definition is the ubiquity of objects having these properties, like the ones in Section 1.1. When making a definition like that of a ring (or group or vector space), the goal is to arrive at a set of axioms that exactly captures the crucial unifying properties of those objects that you wish to study. In familiar number sys- temsliketheintegers,therationalnumbersandtherealnumbers,weareall used to the fact with which Theorem 1.2.2 is concerned, namely that “mul- tiplying by zero gives zero”. The same fact is easily observed to hold in the polynomial ring Q[x] and in the ring of matrices M (R). We might well 2 speculatethatinanyring,itisprobablythecasethatmultiplyingbythezero element always results in the zero element. But before we can assume that this property holds in every ring and incorporate it into our mental scheme for thinking about rings we must deduce this property as a consequence of the ringaxioms. If we were unable to do this, but we only wanted to study rings with the propertydescribedinTheorem1.2.2,wecouldanextraaxiomtoourdefini- tion of a ring insisting on this “multiplication by zero” property. However the fact that this property does turn out to follow from the standard ring axiomsmeansthatitdoesnotneedtobeincludedinthedefinition. 3. On looking at Definition 1.2.1, you may wonder why these seven axioms in particular are chosen to comprise the definition of a ring. Does it look likeanarbitraryselectionofrules? Whydoweinsistthattheadditionhave an identity element and that every element have an inverse for addition, but where the multiplication is concerned only ask that it be associative? Whathappensifweaddmoreaxiomsabouthowthemultiplicationshould behave,ordropsomeoftheaxiomsaboutaddition? Theansweristhatpeo- ple do these things and they lead to different areas of study within abstract algebra. Relaxing the addition axioms in various ways leads to different types of algebraic structures such as near–rings and semirings. If you drop therequirementthatmultiplicationmustbeassociativethenyouarestudy- ing non-associative rings – people do study all of these variants and some of them have important connections to other areas of mathematics. You can even relax the distributive laws and people do this too. However rings themselves as defined in Definition 1.2.1 are of paramount importance in mathematics. On the other hand, if you want more instead of fewer axioms, you can in- sist that multiplication be commutative as well as associative, then you are studying commutative rings. In fact much of this course will be concerned with commutative rings. If you further insist that you want an identity element for multiplication and that every (non-zero) element have an in- 6 verseformultiplication,thenyouarestudyingfields. Fieldsareexamplesof rings,andfieldtheoryitselfisavastareaofmathematicalactivity. Acrucial practice in studying abstract algebra is to be absolutely clear on the precise axiomsthatdeterminetheclassofobjectsthatyouarestudying. 7 1.3 Units in Rings As we have already mentioned, the axioms of a ring are not very restrictive con- cerning how the operation of multiplication should behave - all we ask is that it should be associative. We do not even insist that every ring should contain an identity element for multiplication (although incidentally some authors in ring theory do). If a ring does contain an identity element for multiplication, then we can enter a discussion about whether or not something like division is possible in the ring; we can try to identify pairs of elements that are related to each other in the way that a rational number is related to its reciprocal or in the way that a non-singularmatrixisrelatedtoitsinverse. Definition1.3.1 LetRbearingwithidentityelement1 formultiplication. Anelement R r ∈ RiscalledaunitinRifthereexistss ∈ Rforwhich r×s = 1 ands×r = 1 . R R Inthiscaserandsare(multiplicative)inversesofeachother. Example1.3.2 Q 1. In every element except 0 is a unit; the inverse of a non-zero rational numberisitsreciprocal. 2. In Z the only units are 1 and −1 : no other integer can be multiplied by an integertogive1. 3. In M (R), the units are the 2 × 2 matrices with non-zero determinant, and 2 (cid:18) (cid:19) 1 0 theidentityelementis . 0 1 4. InZ/6Ztheonlyunitsare1¯ and5¯;eachoftheseisitsowninverse. 5. Question for discussion in the seminar : what are the units in M (Z), the ring 2 of2×2matriceswithintegerentries? NOTATION: WewilldenotethesetofunitsinaringRwithidentitybyU(R). REMARKS 1. If R is a unital ring having two or more elements then it follows from The- orem 1.2.2 that the zero element of R and the multiplicative identity in R cannotbethesameelement. 2. If R has two or more elements then 0 cannot be a unit in R, again by Theo- R rem1.2.2. 8 3. It is possible for a ring to have only one element; for example the subset of Z containingonly0isaring. (Thisiscalledthezeroringandasanexample ofaringitisnotveryinstructive) 4. 1 isalwaysaunitinRsinceitisitsowninverse. R The next theorem is concerned with a special property of the subset of a ring consisting of the units. Suppose that R is a unital ring. Then from the above comments it follows that U(R) is a subset of R that includes the (multiplicative) identity element but not the zero element. Is U(R) just a set, or does it have alge- braicstructureofitsown? ThefullringRhasadditionandmultiplicationdefined on it. If we take two units of R we can add them in R; will the result be a unit? If we take two units of R and multiply them (in R), will the result be a unit? If the answer to this second question is yes, then the set of units of R is itself an algebraic structure with respect to the multiplication of R, and we can study its properties. Algebraists are always on the lookout for substructures of the objects that they are studying, which are themselves algebraic structures with respect to the op- eration(s) of the larger object. The general thinking behind this practice is that small things are usually easier to understand than big things, and that we have some chance of understanding (at least partically) a large complicated algebraic structureifwecanidentifysmallerpartsofitthatarethemselvesalgebraicstruc- tures. Theorem1.3.3 Let R be a ring with identity element 1 . Then U(R) is a group under R themultiplicationofR. (U(R)iscalledtheunitgroupofR). Note: ThestatementthatU(R)isagroupundermultiplicationmeansthat: • U(R) is closed under multiplication - whenever elements a and b belong to U(R),sodoestheirproductab. • U(R)containsanidentityelementformultiplication. • U(R)containsamultiplicativeinverseforeachofitselements. ProofofTheorem1.3.3: Weneedtoshow 1. U(R) is closed under the multiplication of R; i.e. that rs is a unit in R when- everrandsareunitsinR. SoassumethatrandsbelongtoU(R)andletr−1 ands−1 denotetheirrespectiveinversesinR. Then (rs)(s−1r−1) = r(ss−1)r−1 = r1 r−1 R = rr−1 = 1 . R 9 Similarly (s−1r−1)(rs) = 1 and so s−1r−1 is an inverse in R for rs, and rs ∈ R U(R). 2. U(R) contains an identity element for multiplication. This is true since 1 ∈ R U(R). 3. U(R)containsaninverseforeachofitselements. To see this, suppose r ∈ U(R), and let r−1 be the inverse of r in R. Then r−1r = 1 andrr−1 = 1 ,soristheinverseofr−1,andr−1 isinU(R). R R (cid:3) Thisprovesthetheorem. EXAMPLES 1. U(Z) = {−1,1}isacyclicgroupoforder2. 2. TheunitgroupofthematrixringM (R)isthegenerallineargroupGL(n,R) n ofn×ninvertiblematricesoverR. 3. The unit group of Q is denoted Q× and consists of all non-zero rational numbers. QUESTION FOR DISCUSSION IN THE SEMINAR: Ingeneral,isthereanythingtobesaid aboutthebehaviourofU(R)withrespecttoadditioninR? Suppose that R is a ring with identity. Then we know that the unit group of R cannot include the zero element of R, but any non-zero element of R could potentially be a unit. A particularly nice thing to happen is for every non-zero elementofRtobeaunit. Ringsinwhichthisoccursareworthyofspecialstudy. Definition1.3.4 A ring with identity is called a field if it is commutative and every non-zeroelementisaunit(sowecandividebyeverynon-zeroelement). ExamplesoffieldsincludeQ,R,CandZ/5Z(check). A ring with identity in which every non-zero element is a unit is called a divi- sion ring. Commutative division rings are fields. Examples of non-commutative divisionringsarenoteasytofind,butwewillseeatleastoneinthiscourse. 10