Rings

Tom Judson · GFDL · Abstract Algebra: Theory and Applications, Ch. 6

A ring is a set with two operations: addition (which forms an abelian group) and multiplication (which is associative and distributes over addition). Rings generalize the integers. Ideals are to rings what normal subgroups are to groups.

Two operations

A ring (R, +, *) satisfies: (1) (R, +) is an abelian group with identity 0; (2) multiplication is associative: a*(b*c) = (a*b)*c; (3) distributive laws: a*(b+c) = a*b + a*c and (a+b)*c = a*c + b*c. Multiplication need not be commutative, and multiplicative inverses may not exist.

Scheme

; Z6 as a ring: addition mod 6, multiplication mod 6

(define (add a b) (modulo (+ a b) 6))
(define (mul a b) (modulo (* a b) 6))

; Distributive law: a * (b + c) = a*b + a*c
(define a 3) (define b 4) (define c 5)

(display "a*(b+c) = ") (display (mul a (add b c))) (newline)
(display "a*b+a*c = ") (display (add (mul a b) (mul a c))) (newline)

; Check: 3*(4+5 mod 6) = 3*3 mod 6 = 9 mod 6 = 3
;        3*4+3*5 mod 6 = 12+15 mod 6 = 0+3 mod 6 = 3
(display "Equal? ") (display (= (mul a (add b c))
                                (add (mul a b) (mul a c))))

Ideals

An ideal I of a ring R is a subring that absorbs multiplication: for any r in R and a in I, both r*a and a*r are in I. Ideals let you build quotient rings R/I, just as normal subgroups let you build quotient groups.

Scheme

; In Z, the even numbers form an ideal.
; For any integer r and even number a, r*a is even.

(define (even? n) (= 0 (modulo n 2)))

; Check absorption: r * a is in ideal for various r, a
(display "Ideal test (multiples of 2 in Z):") (newline)
(let loop ((r -3))
  (when (<= r 3)
    (let ((a 4))  ; a is in ideal (even)
      (display "  ") (display r) (display " * ") (display a)
      (display " = ") (display (* r a))
      (display (if (even? (* r a)) " (even)" " ERROR"))
      (newline))
    (loop (+ r 1))))
; Z/2Z = (even, odd) is the quotient ring.
; It is isomorphic to Z2.

Ring homomorphisms

A ring homomorphism f: R → S preserves both operations: f(a+b) = f(a)+f(b) and f(a*b) = f(a)*f(b). The kernel is an ideal, not just a subgroup.

Scheme

; Ring homomorphism: f: Z -> Z6, f(n) = n mod 6
; Preserves both + and *

(define (f n) (modulo n 6))

(define a 8) (define b 15)

(display "f(a+b) = ") (display (f (+ a b)))
(display "  f(a)+f(b) mod 6 = ") (display (modulo (+ (f a) (f b)) 6)) (newline)

(display "f(a*b) = ") (display (f (* a b)))
(display "  f(a)*f(b) mod 6 = ") (display (modulo (* (f a) (f b)) 6)) (newline)

; Kernel = multiples of 6 = ideal 6Z
(display "Kernel: multiples of 6")

Notation reference

Math	Scheme	Python	Meaning
(R, +, ·)	(add a b) (mul a b)	add(a,b), mul(a,b)	Ring with two operations
I &lhd; R	ideal test	ideal test	I is an ideal of R
R/I	quotient ring	quotient ring	Quotient ring
a(b+c)=ab+ac	verified above	verified above	Distributive law

Neighbors

Algebra track

← Ch.5 Cosets and Lagrange's Theorem — ideals generalize normal subgroups
Ch.7 Integral Domains and Fields — rings with extra properties

Category theory connections

Milewski Ch.6 — types form a semiring under sum and product types (coproduct and product)

Translation notes

Judson treats rings in Ch. 16 and ideals in Ch. 16-17. We compress this because the definition is straightforward: a ring is an abelian group with a second associative operation that distributes. The integers are the motivating example. Commutative rings with identity (the common case in algebra) are assumed throughout. The connection to Milewski's sum/product types is informal: types under (+, *) satisfy distributivity but lack additive inverses, making them a semiring rather than a ring.

← Cosets and Lagrange's Theorem by june.kim Integral Domains and Fields · 7 of 8 →