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A Review of Commutative Ring Theory Mathematics Undergraduate Seminar: Toric Varieties

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A REVIEW OF COMMUTATIVE RING THEORY MATHEMATICS UNDERGRADUATE SEMINAR: TORIC VARIETIES ADRIANO FERNANDES Contents 1. Basic Definitions and Examples 1 2. Ideals and Quotient Rings 3 3. Properties and Types of Ideals 5 4. C-algebras 7 References 7 1. Basic Definitions and Examples In this first section, I define a ring and give some relevant examples of rings we have encountered before (and might have not thought of as abstract algebraic structures.) I will not cover many of the intermediate structures arising between rings and fields (e.g. integral domains, unique factorization domains, etc.) The interested reader is referred to Dummit and Foote. Definition 1.1 (Rings). The algebraic structure “ring” R is a set with two binary opera- tions + and , respectively named addition and multiplication, satisfying · (R, +) is an abelian group (i.e. a group with commutative addition), • is associative (i.e. a, b, c R, (a b) c = a (b c)) , • and the distributive8 law holds2 (i.e.· a,· b, c ·R, (·a + b) c = a c + b c, a (b + c)= • a b + a c.) 8 2 · · · · · · Moreover, the ring is commutative if multiplication is commutative. The ring has an identity, conventionally denoted 1, if there exists an element 1 R s.t. a R, 1 a = a 1=a. 2 8 2 · ·From now on, all rings considered will be commutative rings (after all, this is a review of commutative ring theory...) Since we will be talking substantially about the complex field C, let us recall the definition of such structure. Definition 1.2 (Fields). A commutative ring in which every nonzero element has a mul- tiplicative inverse is called a field, i.e. if a R there exists b Rs.t.a b = 1. 8 2 2 · Date:02/10/2016. 1 2ADRIANOFERNANDES It is trivial to see that the integers form a ring, that integers in modular arithmetic form a ring, and that functions spaces (with point-wise addition and multiplication) are rings. The rational, real and complex numbers are fields. The concept of subsets of a set gives rise to a natural extension to a subring of a ring. Definition 1.3 (Subring). A subring S of a ring R is any subset of R in which the operations of addition and multiplication defined in R make S a ring when restricted to S. Hence, a subring is a ring in itself sitting inside another ring. Transitivity is trivial by the definition. Example 1.4. Z 2Z 4Z 2nZ , for n Z,n>2. This example also shows we can have⊃ an infinite⊃ chain⊃ ··· of⊃ subrings⊃ ··· inside a ring.2 The rings we will be mostly interested in for this seminar will be Polynomial Rings. Definition 1.5 (Polynomial Rings). Fix a commutative ring R with identity. Let x be an indeterminate. The polynomial ring R[x] (read R adjoined x) is the set of all formal n i sums of the form i=0 aix for some n 0 and each ai R. The formal sum is called a polynomial in x with coefficients in the ring≥ R. 2 P If an = 0, then the polynomial is said to have degree n. The polynomial is monic if the coe6 fficient of the largest power is the multiplicative unit. Addition and multi- n i plication are defined component-wise in polynomial rings. Let p(x)= i=0 aix and n i n i q(x)= bix be two polynomials. Then, p(x) q(x):= (ai+bi)x and p(x)+q(x):= i=0 · i=0 P n [ k (a b )]xk. k=0 Pi=0 i k i P The ring over− which the polynomials are formed make a substantial di↵erence, as the followingP P example shows. Example 1.6. Consider p(x)=x+1. If p(x) Z[x], 2p(x)=2x+2 and p(x)2 = x2+2x+1. 2 If p(x) Z/2Z[x], 2p(x) = 0, and p(x)2 = x2 + 1. 2 Before proving some essential facts about polynomial rings, we need some more defini- tions. Definition 1.7 (Integral Domain). An integral domain is any commutative ring R with identity in which there are no zero divisors, i.e. a, b R, a, b =0,ab= 0. 8 2 6 6 Definition 1.8 (Units of a Ring). An element a R is called a unit of the ring if there exists an element b R such that ab = 1. The units2 of a ring form a ring in itself, called 2 the Ring of Units, denoted R⇥ Now we are ready to state an prove some key facts about polynomial rings. Proposition 1.9. Let R be an integral domain, and let p(x),q(x) =0,p(x),q(x) R[x]. Then, 6 2 (1) deg(pq)=deg(p)+deg(q) (2) The units of R and R[x] are the same (3) R[x] is an integral domain. AREVIEWOFCOMMUTATIVERINGTHEORYMATHEMATICSUNDERGRADUATESEMINAR:TORICVARIETIES3 n m m Proof. If p(x) and q(x) have leading terms anx and b x respectively, the leading term n+m of pq(x)isanbmx .SinceR has no zero divisors, anbm = 0. Hence the (3) and (1) hold. For (2), suppose pq(x)=1 deg(p)+deg(q)=0 deg(6p)=deg(q) = 0, hence both p, q ) ) are elements of R and thus are units in R. ⇤ One can establish operation-preserving mappings between rings. We now focus on such mappings. Definition 1.10 (Ring Homomorphism). Let R and S be rings. (1) A ring homomorphism is a map φ : R S satisfying φ(a + b)=φ(a)+φ(b) and φ(ab)=φ(a)φ(b), a, b R 7! (2) The kernel of a ring8 homomorphism2 φ is the set of elements in R mapping to the addition identity of S,i.e.kerφ = r R φ(r)=0S (3) A ring isomorphism is a bijective ring{ 2 homomorphism.| } Example 1.11. Let φn : C[x] Cbedefinedasφ(p(x)) = p(0). This maps a polynomial to its constant term. Since addition7! and multiplication of the constant terms are clearly preserved under this mapping, this is a ring homomorphism. The kernel is the set of all polynomials with constant term 0 in the complex field. Proposition 1.12. Let φ : R S be a ring homomorphism. Then, 7! (1) The image of φ is a subring of S; (2) kerφ is a subring of R and is closed under multiplication by elements of R. Proof. For (1), if s ,s imφ s = φ(r ) ,s = φ(r ) for some r ,r R.Sinceφ is a 1 2 2 ) 1 1 2 2 1 2 2 homomorphism, s1 s2 = φ(r1 r2) and s1s2 = φ(r1r2), so s1 s2 ,s1s2 S. For (2), if a, b kerφ φ(−a)=φ(b)=0− φ(ab)=φ(a b) = 0, so the− kernel is2 closed under subtraction2 and) multiplication. Now,) for any r −R, φ(ra)=φ(r)φ(a)=φ(r)0 = 0 = 0φ(r)=φ(a)φ(r)=φ(ar), so indeed the kernel is2 closed under multiplication by elements of R. ⇤ Kernels are examples of structures appearing more generally in rings, so called ideals. We now focus on these structures. 2. Ideals and Quotient Rings Definition 2.1 (Ideals). Let R be a ring, r R and I R. I is an ideal of R if I is a subring of R that is closed under multiplication2 by the elements⇢ of R,i.e.ifa I,then ra, ar I; one writes rI I r R. 2 2 ⇢ 8 2 It turns out that there is a vital connection between ideals and ring homomorphisms. Ideals are kernels for some ring homomorphism of the ring in which they sit in. The cosets of any element of R,sayr+I, r R, I an ideal in R form a ring in itself if we appropriately define addition and multiplication2 of cosets for the set of cosets. Ideals are in fact what we 4ADRIANOFERNANDES require for those operations as we here present to make the set of cosets into a ring. We define addition and multiplication for cosets of elements of R by an ideal I as follows: (r + I)+(s + I)=r + s + I (r + I)(s + I)=rs + I It is trivial to show that the distributive law holds whence operations are defined as above. Let r, s, t R and I an ideal in R.Then 2 (r + I)[(s + I)+(t + I)] = (r + I)[s + t + I] = r(s + t)+I =(rs + rt)+I =(rs + I)+(rt + I)=(r + I)(s + I)+(r + I)(t + I) So it seems that we can indeed form a ring by considering the cosets of elements of a ring by the ideals of a ring. The discussion above proved the following Proposition 2.2. Let R be a ring and I an ideal. If addition and multiplication are defined as above, the set of cosets of elements of R by the ideal I form a ring. Conversely, any subgroup I of R in which the above operations are well-defined form an ideal of R. I skip the proof of the converse direction for succinctness. Definition 2.3 (Quotient Rings). Let R be a ring and I be an ideal of it. The ring R/I with operations defined above form the quotient ring of R, whose elements of the form r + I for r R. 2 The following theorem summarizes the present discussion: Theorem 2.4 (The First Ring Isomorphism Theorem).
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