{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture24 - 24 Rings 24.1 Definitions and basic examples...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
24. Rings 24.1. Definitions and basic examples. Definition. A ring R is a set with two binary operations + (addition) and · (multiplication) satisfying the following axioms: (A0) R is closed under addition (A1) addition is associative (A2) there exists 0 R s.t. a + 0 = 0 + a for all a R (A3) for every a R there exists - a R s.t. a + ( - a ) = ( - a ) + a = 0 (A4) addition is commutative (M0) R is closed under multiplication (M1) multiplication is associative (D1) distributivity on the left: ( a + b ) c = ac + bc for all a, b, c R (D2) distributivity on the right: c ( a + b ) = ca + cb for all a, b, c R Remark: The axioms (A0)-(A3) simply say that R is a group with respect to addition. Axiom (A4) says that the group ( R, +) is abelian. Definition. A ring R is called commutative if multiplication is commuta- tive, that is, ab = ba for all a, b R . Definition. A ring R is a called a ring with 1 (or a ring with unity ) if there exists 1 R s.t. a · 1 = 1 · a = a for all a R . Definition. A ring R is called a field if (i) R is commutative, (ii) R is a ring with 1 (iii) for every a R , with a 6 = 0, there exists a - 1 R such that a · a - 1 = a - 1 · a = 1 (iv) 1 6 = 0. Examples: 1. Fields. The examples of fields we know so far are Q , R , C and Z p where p is prime. 2. Commutative rings with 1, which are not fields. The examples we have seen so far are Z and Z n where n is non-prime. Another important class of examples is given by polynomial rings: Let R be a commutative ring with 1, and let R [ x ] denote the set of all polynomials with coefficients in R . One can think of R [ x ] as the set of formal 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2 expressions a 0 + a 1 x + . . . + a n x n , with a i R . Addition and multiplication on R [ x ] are defined according to the “usual algebra rules”, e.g. ( a 0 + a 1 x ) + ( b 0 + b 1 x + b 2 x 2 ) = ( a 0 + b 0 ) + ( a 1 + b 1 ) x + b 2 x 2 and ( a 0 + a 1 x ) · ( b 0 + b 1 x + b 2 x 2 ) = a 0 b 0 +( a 0 b 1 + a 1 b 0 ) x +( a 0 b 2 + a 1 b 1 ) x 2 + a 1 b 2 x 3 . It is a rather long but straightforward calculation to check that R [ x ] with these operations becomes a ring with 1. The unity element of R [ x ] is the constant polynomial 1 (where 1 is the unity element of R ). 3. Commutative rings without 1. The basic examples are the rings n Z where n 2 is a fixed integer. 4. Noncommutative rings. The basic examples are the matrix rings Mat n ( F ) where F is some field and n 2.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}