Relating Different Algebraic Structures: One Binary Operation

Home , Algebra over a field

Algebraic structures A group is a set G with a binary operation satisfying associativity, and the ˚ ‹ existence of an identity and inverses. A monoid is a set M with a binary operation satisfying associativity and ˚ ‹ the existence of an identity. (e.g. Z, ) p ˆq A semigroup is a set S with a binary operation satisfying associativity (e.g. ˚ ‹ Z 0, ). p ° `q A field is a set F with two binary operations, and ,suchthat F, and ˚ ` ˆ p `q F 0 , are abelian groups, and the distributive property holds between p ´t u ˆq and .(e.g.R or Z pZ) ` ˆ { A vector space (over a field F )isaanabeliangroup V, that has a ˚ p `q faithful action on it by F that satisfies the distributive properties. (i.e. for n f F , u, v V , f u v f u f v). (e.g. F R, V R ) P P ¨p ` q“ ¨ ` ¨ “ “ A ring (with identity) is a set R with two binary operations, and ,such ˚ ` ˆ that R, is an abelian group, and R, is a monoid, and the distributive p `q p ˆq property holds. (e.g. Z or Z x ) r s An algebra (over a field F )isaringA such that A, forms a vector space ˚ p `q over F and the distributive properties hold. (e.g. F R, A R x ) “ “ r s A module (over a ring R)isanabeliangroup M, with an action from R ˚ p `q that satisfies the distributive properties. (e.g. R Z acts on M Z nZ by “ “ { multiplication) ...and many many others.

Relating di↵erent algebraic structures: one binary operation

A semigroup is a set S with a binary operation satisfying ‹ associativity. A monoid is a semigroup M, that has an identity element. p ‹q A group is a monoid G, that has an inverse for every element. p ‹q

groups monoids semigroups t u Ä t u Ä t u Relating di↵erent algebraic structures: two binary operations

A ring (with identity) is a set R with two binary operations, ` and ,suchthat R, is an abelian group, and R, is a ˆ p `q p ˆq monoid, and the distributive property holds. A ﬁeld is a ring R where R 0 , is an abelian group. p ´t u ˆq An algebra (over a ﬁeld F )isaringA such that A, forms a p `q vector space over F and the distributive properties hold.

ﬁelds rings w/ 1 groups t u Ä t u Ä t u

algebras rings w/ 1 groups t u Ä t u Ä t u

Relating di↵erent algebraic structures: actions

A vector space (over a field F ) is a an abelian group V, that has a faithful action on it by F that satisfies the distributivep properties.`q An algebra (over a field F )isaringA such that A, forms a vector space over F and the distributive properties hold.p `q A module (over a ring R) is an abelian group M, with an action from R that satisfies the distributive properties.p `q

algebras vector spaces modules groups t u Ä t u Ä t u Ä t u PART II: RING THEORY A ring R is a set together with two binary operations and such that ` ˆ (a) R, is an abelian group, (b) p is`q associative: a b c a b c , ˆ p ˆ qˆ “ ˆp ˆ q (i.e. R, is a semigroup) p ˆq (c) the distributive laws hold for R

a b c a c b c and a b c a b a c . p ` qˆ “p ˆ q`p ˆ q ˆp ` q“p ˆ q`p ˆ q The ring R is said to have an identity (R is a ring with identity)if there is an element 1 R with P 1 a a 1 a for all a R. ˆ “ ˆ “ P (i.e. R, is a monoid) p ˆq AringR is a division ring if R 0 , is a group. p ´t u ˆq (i.e. R has 1 and multiplicative inverses) AringR is commutative if R, is commutative. p ˆq A commutative division ring is a ﬁeld. We usually write ab instead of a b. ˆ

Some favorite examples 1. Since Z pZ, Q, R,andC are all fields, they hare also rings, { rings with 1, commutative rings, and division rings. 2. Z and Z nZ are commutative rings with 1, but not a division { rings. 3. Mn F n n matrices with entries in the field F is a p q“t ˆ u ring with 1. It is not commutative, nor is it a division ring. In fact, Mn R is a ring for R aringaswell. p q 4. If R is a commutative ring, then so are n polynomials: R x r0 r1x rnx ri R, n Z 0 , r s“t ` `¨¨¨` | P P • u power series: R x r r x r R , rr ss “ t 0 ` 1 `¨¨¨ | i P u Laurent polynomials: 1 m m 1 n R x, x´ rmx rm 1x ` rnx ri R, m n Z , r s“t ` ` `¨¨¨` | P § P u m m 1 Laurent series: R x rmx rm 1x ` ri R, m Z . pp qq “ t ` ` `¨¨¨ | P P u Tip: For every new definition or theorem, ask yourself what it means for

R, Z, Z 3Z, Z 6Z,M2 R , R x , Z x , and Z 6Z x . { { p q r s r s { r s Some favorite examples

3. For a ﬁeld F , M F n n matrices with entries in F . np q“t ˆ u 4. For a commutative ring R, n R x r0 r1x rnx ri R, n Z 0 . r s“t ` `¨¨¨` | P P • u

Note that Mn F and R x are both rings whose elements are functions. In both examples,p q the binaryr s operation is function addition: ` M1 M2 v M1 v M2 v and f g x f x g x . (Okp because` theqp imagesq“ ofp q` the functionsp q arep additive` qp q“ groups.)p q` p q However, the binary operation playing the role of is fundamentally di↵erent between the two examples! ˆ In M F , “multiplication” is function composition: np q matrix multiplication is the same as linear function composition, i.e. M M v M M v . p 1 2qp q“ 1p 2p qq But in R x , “multiplication” is function multiplication: r s the image of the product is the product of the images, i.e. fg x f x g x . (Can’t do this unless the imagep qp of theq“ functionsp q p q is a ring!)

Rings of functions In general, let A be a ring, and let X be a set. Then the set of functions R f : X A is a ring under the binary operations “t Ñ u f g x f x g x and fg x f x g x . ( ) p ` qp q“ p q` p q p qp q“ p q p q ˚ Putting restrictions on the kinds of functions in your set can sometimes also produce rings (e.g. polynomial functions). Example: Which of the following form rings under the binomial operations given in ( )? Why or why not? ˚ (a) di↵erentiable functions R R (f 1 x is deﬁned x R) t Ñ u p q @ P (b) even functions R R (f x f x x R) t Ñ u p´ q“ p q@ P (c) odd functions R R (f x f x x R) t Ñ u p´ q“´ p q@ P (d) strictly positive functions R R (f x 0 x R) t Ñ u p q ° @ P (e) non-zero functions R R (f x 0 x R) t Ñ u p q‰ @ P A subring of the a R is a subgroup of R that is closed under multiplication. Think: what are some subrings of our fav examples? Subring criterion: S R is a subring i↵ S ,andS is closed Ñ ‰H under subtraction and multiplication. Mixing the two binary operations

As usual, we denote 0 is the additive identity, 1 is the multiplicative identity (if it exists), a is the additive inverse to a,and ´ 1 a´ is the multiplicative inverse to a (if it exists). Proposition Let R be a ring. 1. 0a a0 0 for all a R. “ “ P 2. a b a b ab for all a, b R. p´ q “ p´ q“´p q P 3. a b ab for all a, b R. p´ qp´ q“ P 4. If R has an identity 1, the identity is unique and a 1 a. ´ “p´ q Deﬁnition Let R be a ring. 1. A nonzero element a R is called a zero divisor if there is a P nonzero element b R such that ab 0 or ba 0. P “ “ Think: what are some zero divisors of our fav examples? 2. Assume R has identity 1 0.Anelementu R is called a ‰ P unit in R if there is some v R such that uv vu 1. P “ “ The set of units R is a group (under )calledgroup of ˆ ˆ units of R. Think: what are the groups of units in our fav examples? 3. A commutative ring with identity 1 0 is called an integral ‰ domain if it has no zero divisors. Think: which our fav examples are integral domains?

Proposition Let R be a ring and a, b, c R such that a is not a zero divisor. P If ab ac,thena 0 or b c. (“zero divisors are bad for cancellation”) “ “ “ If particular, if R is an integral domain and ab ac, “ then a 0 or b c. (“cancellation always works in integral domains”) “ “ Corollary Any ﬁnite integral domain is a ﬁeld. More on polynomial rings

Deﬁnition Let R be a commutative ring with identity. The formal sum

n n 1 anx an 1x ´ a1x a0 ` ´ `¨¨¨` `

with n 0 and each ai R is called a polynomial in x with • P coecients ai in R. n If an 0, the polynomial is of degree n, anx is the leading term ‰ and an is the leading coecient. The polynomial is monic if an 1. “ The set of all such polynomials is the ring of polynomials in the x with coecients in R, denoted R x .TheringR appears in R x r s r s as the constant polynomials.