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Linear Algebra's Basic Concepts

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Linear Algebra's Basic Concepts Algebraic Terminology UCSD Math 15A, CSE 20 (S. Gill Williamson) Contents > Semigroup @ ? Monoid A @ Group A A Ring and Field A B Ideal B C Integral domain C D Euclidean domain D E Module E F Vector space and algebra E >= Notational conventions F > Semigroup We use the notation = 1; 2;:::; for the positive integers. Let 0 = 0 1 2 denote the nonnegative integers, and let 0 ±1 ±2 de- ; ; ;:::; { } = ; ; ;::: note the set of all integers. Let × n ( -fold Cartesian product of ) be the set { } S n { S } of n-tuples from a nonempty set S. We also write Sn for this product. Semigroup!Monoid!Group: a set with one binary operation A function w : S2 ! S is called a binary operation. It is sometimes useful to write w(x; y) in a simpler form such as x w y or simply x · y or even just x y. To tie the binary operation w to S explicitly, we write (S; w) or (S; ·). DeVnition >.> (Semigroup). Let (S; ·) be a nonempty set S with a binary oper- ation “·” . If (x · y) · z = x · (y · z), for all x, y, z 2 S, then the binary operation “·” is called associative and (S; ·) is called a semigroup. If two elements, s; t 2 S satisfy s · t = t · s then we say s and t commute. If for all x; y 2 S we have x · y = y · x then (S; ·) is a commutative (or abelian) semigroup. Remark >.? (Semigroup). Let S = M2;2(e) be the set of 2 × 2 matrices with en- tries in e = 0; ±2; ±4;::: , the set of even integers. DeVne w(X;Y ) = XY to be the standard multiplication of matrices (which is associative). Then { } (S; w) is a semigroup. This semigroup is not commutative. The semigroup of even integers, (e; ·), where “·” denotes multiplication of integers, is commutative. @ Associative + Identity = Monoid ? Monoid DeVnition ?.> (Monoid). Let (S; ·) be a semigroup. If there exists an element e 2 S such that for all x 2 S, e · x = x · e = x, then e is called an identity for the semigroup. A semigroup with an identity is called a monoid. If x 2 S and there is a y 2 S such that x · y = y · x = e then y is called an inverse of x. Remark ?.? (Monoid). The identity is unique (i.e., if e and e0 are both identities then e = e · e0 = e0). Likewise, if y and y0 are inverses of x, then y0 = y0 · e = y0 · (x · y) = (y0 · x) · y = e · y = y so the inverse of x is unique. The 2 × 2 matrices, M2;2(), with matrix multiplication form a monoid (identity I2, the 2 × 2 identity matrix). Associative + Identity + Inverses = Group @ Group DeVnition @.> (Group). Let (S; ·) be a monoid with identity e and let x 2 S. If there is a y 2 S such that x · y = y · x = e then y is called an inverse of x (see ?.>). A monoid in which every element has an inverse is a group. Remark @.? (Group). The mathematical study of groups is a vast subject. Commutative groups, x · y = y · x for all x and y, play an important role. They are also called abelian groups. Note that the inverse of an element x in a group is unique: if y and y0 are inverses of x, then y0 = y0 · e = y0 · (x · y) = (y0 · x) · y = e · y = y (see ?.?). Ring: one set with two intertwined binary operations A Ring and Field DeVnition A.> (Ring and Field). A ring, (S; +; ·), is a set with two binary operations such that (S; +) is an abelian group (“ + " is called “addition”) and (S − 0 ; ·) is a semigroup (“ · " is called “multiplication”). The two operations are related by distributive rules which state that for all , , in : { } x y z S (left) z · (x + y) = z · x + z · y and (x + y) · z = x · z + y · z (right): A Remark A.? (Notation and special rings). The identity of the abelian group (S; +) is denoted by “0” and is called the zero of the ring (S; +; ·). If (S − 0 ; ·) is a monoid then + · is a ring with identity. If − 0 · is commutative (S; ; ) (S ; ) { } then + · is a commutative ring. If − 0 · is a group then the ring is (S; ; ) (S ; ) { } called a skew-Veld or division ring. If this group is abelian then the ring is { } called a Veld. Remark A.@ (Basic ring identities). We have taken the point of view that a semigroup, (S; ·), has S nonempty (>.>). Thus, the semigroup (S − 0 ; ·) of a ring (A.>) has − 0 nonempty. If are in a ring + · then the S r; s;t (S; ; ) { } following basic identities (in braces, plus hints for proof) hold: { } (1) r · 0 = 0 · r = 0 : If x + x = x then x = 0. Take x = r · 0 and x = 0 · r. (2) {(−r) · s = r · (−}s) = −(r · s) : r · s + (−r) · s = 0 =) (−r) · s = −(r · s). (3) {(−r) · (−s) = r · s : Replace} r by −r in (?). Note that −(−r) = r. Using{ the identities of} Remark A.@, you can show that if (S − 0 ; ·) has an identity , then − · − for any 2 and, taking − , − · − . e ( e) a = a a S a = e ( {e}) ( e) = e It is convenient to deVne r − s = r + (−s). Then we have t · (r − s) = t · r − t · s and (r − s) · t = r · t − s · t. A Veld is a ring (S; +; ·) where (S − 0 ; ·) is an abelian group { } Remark A.A (Ring and Field). The 2 × 2 matrices over the even integers, M2;2(e), with the usual multiplication and addition of matrices, is a non- commutative ring without an identity. The matrices, M2;2(), over all inte- gers, is a noncommutative ring with identity. The ring of 2 × 2 matrices of ! x y the form where x and y are complex numbers is a skew-Veld but −y x not a Veld. This skew-Veld is equivalent to (i.e, a “matrix representation of”) the skew Veld of quaternions (see Wikipedia article on quaternions). The most important Velds for us will be the Velds of real and complex numbers. B Ideal DeVnition B.> (Ideal). Let (R; +; ·) be a ring and let A ⊆ R be a nonempty subset of R. If (A; +; ·) is a ring, then it is called a subring of (R; +; ·).A subring (A; +; ·) is a left ideal if for every x 2 R and y 2 A, xy 2 A.A right ideal is similarly deVned. If (A; +; ·) is both a left and right ideal then it is a two-sided ideal or, simply, an ideal. Note that if (R; +; ·) is commutative then all ideals are two sided. B ! Remark B.? (Ideal). The set of all matrices of the form x y is a sub- a = 0 0 ring of M2;2(). This subring, which has no identity element, is a right ideal but not a left ideal. It also has zero divisors - elements a , 0 and b , 0 such that a · b = 0. Another example of an ideal is the set of even integers, e, which is a subring of the ring of integers, (which, it is worth noting, has no zero divisors). The subring, e, is an ideal (two-sided) in . Given any integer n , 0, the set k · n j k 2 of multiples of n, is an ideal of the ring which we denote by . Such an ideal (i.e., generated by a single ({n) = n = }n element, n) in is called a principal ideal. It is easy to see that all ideals in are principal ideals. Another nice property of integers is that they uniquely factor into primes (up to order and sign). C Integral domain Algebraists have deVned a number of important abstractions of the ring of integers, . We next discuss four such abstractions: integral domain, prin- cipal ideal domain (PID), unique factorization domain (UFD), and Euclidean domains - each more restrictive than the other. Euclidean Domain =) PID =) UFD DeVnition C.> (Integral domain). An integral domain is a commutative ring with identity, (R; +; ·), with no zero divisors (B.?). We denote the identity of (R − 0 ; ·) by 1R or, simply, 1. DeVnition{ } C.? (Characteristic of a ring). Let R be a ring. Given a 2 R and an integer n > 0, deVne na ≡ a + a + ··· a where there are n terms in the sum. If there is an integer n > 0 such that na = 0 for all a 2 R then the characteristic of R is the least such n. If no such n exists, then R has characteristic zero. Remark C.@ (Divisors, associates and primes). If b, c and a are elements of an integral domain R such that a , 0 and b = a · c then we say that a divides b (written a j b) or a is a divisor of b. An element u in R − 0 is an invertible element or a unit of if has an inverse in − 0 · . Two elements, R u (R ; ) { } a and , of are associates in if · where is a unit. An element in b R R a = b u u { } p R − 0 is irreducible if p = a · b implies that either a or b is invertible and prime if j · implies j or j . For unique factorization domains (C.A), { } p a b p a p b p is irreducible if and only if it is prime. In the ring , the only invertible elements are +1; −1 . The only associates of an integer n , 0 are +n and − . The integer 12 3 · 4 is the product of two non-invertible elements so n { =} 12 is not irreducible (i.e., reducible) or, equivalently in this case, not a prime.
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