Localization and Classification of Zero-divisor Graphs of Rings Tom Cuchta, Kathryn Lokken, William Young Marshall University, University of Wisconsin, Purdue University Localization and Classification of Zero-divisor Graphs of Rings – p. 1/37 Basic ring theory • A ring R is a set with two operations: + and ·, and it has most of the nice properties that you "would like" an algebraic structure to have. Localization and Classification of Zero-divisor Graphs of Rings – p. 2/37 Basic ring theory • A ring R is a set with two operations: + and ·, and it has most of the nice properties that you "would like" an algebraic structure to have. • In general, rings do not have the property of commutativity on multiplication or a multiplicative identity, but for this talk, all rings have both. Localization and Classification of Zero-divisor Graphs of Rings – p. 2/37 Basic ring theory • A ring R is a set with two operations: + and ·, and it has most of the nice properties that you "would like" an algebraic structure to have. • In general, rings do not have the property of commutativity on multiplication or a multiplicative identity, but for this talk, all rings have both. • Such rings are called commutative rings with identity. Localization and Classification of Zero-divisor Graphs of Rings – p. 2/37 Basic ring theory The best way to learn about rings is to look at some: • The integers, Z, notice no elements have multiplicative inverses other than 1 and −1 Localization and Classification of Zero-divisor Graphs of Rings – p. 3/37 Basic ring theory The best way to learn about rings is to look at some: • The integers, Z, notice no elements have multiplicative inverses other than 1 and −1 • The integers modulo n, Zn, is a ring under addition modulo n and multiplication modulo n Localization and Classification of Zero-divisor Graphs of Rings – p. 3/37 Basic ring theory The best way to learn about rings is to look at some: • The integers, Z, notice no elements have multiplicative inverses other than 1 and −1 • The integers modulo n, Zn, is a ring under addition modulo n and multiplication modulo n • The 2 × 2 matrices with real entries Localization and Classification of Zero-divisor Graphs of Rings – p. 3/37 Basic ring theory The best way to learn about rings is to look at some: • The integers, Z, notice no elements have multiplicative inverses other than 1 and −1 • The integers modulo n, Zn, is a ring under addition modulo n and multiplication modulo n • The 2 × 2 matrices with real entries • Z2[x], polynomials with coefficients from Z2 Localization and Classification of Zero-divisor Graphs of Rings – p. 3/37 Basic ring theory • An element z ∈ R is called a zero-divisor if there is a nonzero z′ ∈ R such that zz′ = 0. The set of zero-divisors is denoted Z(R). Localization and Classification of Zero-divisor Graphs of Rings – p. 4/37 Basic ring theory • An element z ∈ R is called a zero-divisor if there is a nonzero z′ ∈ R such that zz′ = 0. The set of zero-divisors is denoted Z(R). • For example, in Z12, 8 · 3 (mod 12) = 24 (mod 12) ≡ 0 (mod 12), so 8 and 3 are zero-divisors in Z12. Localization and Classification of Zero-divisor Graphs of Rings – p. 4/37 Basic ring theory • An element z ∈ R is called a zero-divisor if there is a nonzero z′ ∈ R such that zz′ = 0. The set of zero-divisors is denoted Z(R). • For example, in Z12, 8 · 3 (mod 12) = 24 (mod 12) ≡ 0 (mod 12), so 8 and 3 are zero-divisors in Z12. • An element u ∈ R is called a unit if there exists an element u−1 ∈ R such that uu−1 = 1. The set of units in R is denoted U(R). Localization and Classification of Zero-divisor Graphs of Rings – p. 4/37 Basic ring theory • A subring of a ring R is a subset of R that is also ring. Localization and Classification of Zero-divisor Graphs of Rings – p. 5/37 Basic ring theory • A subring of a ring R is a subset of R that is also ring. • An ideal, I, of R is an especially nice subring of R such that for every r ∈ R and i ∈ I, ri ∈ I. In a sense, an ideal is a subring that "absorbs" ring multiplication. Localization and Classification of Zero-divisor Graphs of Rings – p. 5/37 Basic ring theory • A subring of a ring R is a subset of R that is also ring. • An ideal, I, of R is an especially nice subring of R such that for every r ∈ R and i ∈ I, ri ∈ I. In a sense, an ideal is a subring that "absorbs" ring multiplication. • A prime ideal, P of R is an ideal of R with the following property: if r, r′ ∈ R such that rr′ ∈ P , then r ∈ P or r′ ∈ P . Localization and Classification of Zero-divisor Graphs of Rings – p. 5/37 Basic ring theory • If a ∈ R, then we define (a) to be the principal ideal generated by a. This is, specifically, the set of multiples of a. Localization and Classification of Zero-divisor Graphs of Rings – p. 6/37 Basic ring theory • If a ∈ R, then we define (a) to be the principal ideal generated by a. This is, specifically, the set of multiples of a. • For example, in Z12, (2) = {0, 2, 4, 6, 8, 10}. Localization and Classification of Zero-divisor Graphs of Rings – p. 6/37 Basic ring theory • If a ∈ R, then we define (a) to be the principal ideal generated by a. This is, specifically, the set of multiples of a. • For example, in Z12, (2) = {0, 2, 4, 6, 8, 10}. • A ring such that all ideals are principal is called a principal ideal ring. Localization and Classification of Zero-divisor Graphs of Rings – p. 6/37 Basic ring theory • If a ∈ R, then we define (a) to be the principal ideal generated by a. This is, specifically, the set of multiples of a. • For example, in Z12, (2) = {0, 2, 4, 6, 8, 10}. • A ring such that all ideals are principal is called a principal ideal ring. • For all n, Zn is a principal ideal ring. The prime ideals are generated by the prime numbers in this ring. Localization and Classification of Zero-divisor Graphs of Rings – p. 6/37 Basic ring theory • The direct product of two rings R1 and R2, denoted R1 × R2, is the set of ordered pairs (r1, r2) such that r1 ∈ R1 and r2 ∈ R2. Localization and Classification of Zero-divisor Graphs of Rings – p. 7/37 Basic ring theory • The direct product of two rings R1 and R2, denoted R1 × R2, is the set of ordered pairs (r1, r2) such that r1 ∈ R1 and r2 ∈ R2. • For example, Z2 × Z2 = {(0, 0), (1, 0), (0, 1), (1, 1)}. Localization and Classification of Zero-divisor Graphs of Rings – p. 7/37 Basic ring theory • The direct product of two rings R1 and R2, denoted R1 × R2, is the set of ordered pairs (r1, r2) such that r1 ∈ R1 and r2 ∈ R2. • For example, Z2 × Z2 = {(0, 0), (1, 0), (0, 1), (1, 1)}. • Operations in direct product rings are componentwise, meaning that (r1, r2) + (r3, r4) = (r1 + r3, r2 + r4) and (r1, r2)(r3, r4) = (r1r3, r2r4). Localization and Classification of Zero-divisor Graphs of Rings – p. 7/37 Basic ring theory • Two rings, R1 and R2, are said to be isomorphic if there exists a one-to-one and ′ onto map, φ : R1 −→ R2 such that if r, r ∈ R1, then φ(r + r′)= φ(r)+ φ(r′) and φ(rr′)= φ(r)φ(r′). Localization and Classification of Zero-divisor Graphs of Rings – p. 8/37 Basic ring theory • Two rings, R1 and R2, are said to be isomorphic if there exists a one-to-one and ′ onto map, φ : R1 −→ R2 such that if r, r ∈ R1, then φ(r + r′)= φ(r)+ φ(r′) and φ(rr′)= φ(r)φ(r′). • For example, there is an isomorphism between the rings Z6 and Z2 × Z3 through the function φ(0) = (0, 0),φ(1) = (1, 1),φ(2) = (0, 2),φ(3) = (1, 0),φ(4) = (0, 1), and φ(5) = (1, 2). Localization and Classification of Zero-divisor Graphs of Rings – p. 8/37 Localizations • Since we know that not all elements in rings have multiplicative inverses, we might want to extend a ring to have multiplicative inverses for some subset. The extended ring is called a localization, and it is comprised of r equivalence classes of fractions, denoted [s]. Localization and Classification of Zero-divisor Graphs of Rings – p. 9/37 Localizations • Since we know that not all elements in rings have multiplicative inverses, we might want to extend a ring to have multiplicative inverses for some subset. The extended ring is called a localization, and it is comprised of r equivalence classes of fractions, denoted [s]. • We will learn that, in fact, you have probably done this before! But first, some more definitions. Localization and Classification of Zero-divisor Graphs of Rings – p. 9/37 Localizations • Since we are dealing with fractions and their equivalence classes, a good place to start is to define the equivalence class we are using. Localization and Classification of Zero-divisor Graphs of Rings – p. 10/37 Localizations • Since we are dealing with fractions and their equivalence classes, a good place to start is to define the equivalence class we are using.