MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 11. Basic definitionsIn the first week, I reviewed the basic definitions and rephrased themin a categorical framework. The purpose of this was two-fold. First, Iwanted to make the preliminaries more interesting for those of you whoalready know the basic concepts. Second, I want students to feel morecomfortable with category theory so that I can use it later to explainmore difficult concepts.1.1. Rings and endomorphisms. First of all, an additive group isdefined to be an abelian group in which the composition law is writtenas addition and the neutral element is called “0”. When the group lawis written as multiplication, it is called a multiplicative group whetheror not it is abelian.Definition 1.1. A ring is an additive group R together with a biad-ditive, asso ciative multiplication law with unity. More precisely, a ringis: (R, +, ·, 0, 1) where(1) (R, +, 0) is an additive group.(2) (R, ·, 1) is a monoid. I.e., multiplication is associative and hasunit 1.(3) Multiplication distributes over addition from the left and theright. I.e.,a(b + c) = ab + ac (a + b)c = ac + bcThis condition is called bi-additivity since the multiplicationmapping (x, y) !→ xy is additive (a homomorphism) in eachvariable.Lang allows 1 = 0. I don’t. But there is only one ring with thisproperty, namely the zero ring, since any ring with 1 = 0 has theproperty thatx = 1x = 0x = 0for all x ∈ R.To make a category we need homomorphisms.Definition 1.2. If R, S are rings, a ring homomorphism φ : R → S isa set mapping which is(1) additive: φ(r + s) = φ(r) + φ(s)(2) multiplicative: φ(rs) = φ(r)φ(s)(3) unital: φ(1) = 1.I pointed out that the image of a ring homomorphism is a subringof S, i.e., a subset of S which is closed under addition, subtraction,2 MATH 101A: ALGEBRA I PART B: RINGS AND MODULESmultiplication and contains 1. The kernel of φ is a (two-sided) idealin R, i.e., a subset I ⊆ R so that I is an additive subgroup of R andRI = I = IR.One example of a ring is given by the endomorphism ring of anynonzero (!) additive group. If A, B are additive groups, then HomAdd(A, B)is also an additive group where addition is defined pointwise:(f + g)(x) = f(x) + g(x).Here Add is the category of additive groups and homomorphisms. Inthe case A = B, homomorphisms f : A → A are called endomorphismsof A and we write End(A) = Hom(A, A). Being a Hom set, it is anadditive group. But now we also have a composition law◦ : End(A) × End(A) → End(A).The composition law distributes over addition on both sides for differ-ent reasons:(1) Distributivity from the left comes from the fact that these arehomomorphisms:f(g + h)(x) = f(g(x) + h(x)) = f g(x) + f h(x) = (f g + f h)(x).(2) Distributivity from the right comes from the definition of addi-tion in End(A):(f + g)h(x) = fh(x) + gh(x) = (f h + gh)(x).Composition of mappings is always associative and the identity map-ping acts as unity id : A → A. Therefore, (EndAdd(A), +, ◦, 0, id) is aring (provided that A '= 0).The idea is that End(A) has addition and multiplication (given bycomposition) satisfying a list of conditions. Rings are subsets R ⊆End(A) which has all of this structure. I.e., R is closed under addition,subtraction, multiplication and contains 0 and 1.Question: If we start with a ring R then what is A?Answer: We can take A = (R, +), the underlying additive group ofR. Then we have a ring monomorphismφ = λ : R → End(R, +)given by φ(r) = λrwhich is left multiplication by r. This is an additiveendomorphism of R by left distributivity:λr(a + b) = r(a + b) = ra + rb = λr(a) + λr(b).The additivity of the mapping φ = λ follows from right distributivity(in R):φ(r+s)(x) = λr+s(x) = (r+s)x = rx+sx = λr(x)+λs(x) = (φ(r)+φ(s))(x).MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 3The fact that φ is multiplicative follows from the associativity of mul-tiplication in R:φ(rs)(x) = (rs)x = r(sx) = φ(r)φ(s)(x).The fact that φ is a monomorphism follows from the fact that 1 is aright unity:(∀x ∈ R )(φ(r)(x) = 0) ⇒ φ(r)(1) = r1 = r = 0.The fact that φ(1) = 1 follows from the fact that 1 is a left unity:φ(1)(x) = 1x = x.Thus, all of the properties of a ring (the ones which involve the multi-plication), are included in the statement that φ = λ : R → End(R, +)is a ring homomorphism.1.2. Modules. When we think of R as being a subring of EndAdd(A),the additive group A is called an R-module.Definition 1.3. An R-module is an additive group M together with aring homomorphism φ : R → EndAdd(M).This is usually stated in longhand as follows. For every r ∈ R, x ∈ Mthere is rx ∈ M with the following prop erties for all r, s ∈ R andx, y ∈ M.(1) r(x + y) = rx + ry (I.e., φ(r) is additive, or equivalently, φ is aset mapping.)(2) (r + s)x = rx + sx (I.e., φ is additive.)(3) (rs)x = r(sx) (I.e., φ is multiplicative.)(4) 1x = x (I.e., φ(1) = id = 1.)For example, R is an R-module and any left ideal in R is an R-module. (A left ideal is a proper additive subgroup I ⊂ R so thatRI =