Chapter4 - Chapter 4 Module Fundamentals 4.1 Modules and...

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Chapter 4 Module Fundamentals 4.1 Modules and Algebras 4.1.1 Defnitions and Comments A vector space M over a feld R is a set oF objects called vectors, which can be added, subtracted and multiplied by scalars (members oF the underlying feld). Thus M is an abelian group under addition, and For each r R and x M we have an element rx M . Scalar multiplication is distributive and associative, and the multiplicative identity oF the feld acts as an identity on vectors. ±ormally, r ( x + y )= + ry ;( r + s ) x = + sx ; r ( sx )=( rs ) x ;1 x = x For all x,y M and r,s R . A module is just a vector space over a ring. The Formal defnition is exactly as above, but we relax the requirement that R be a feld, and instead allow an arbitrary ring. We have written the product with the scalar r on the leFt, and technically we get a left R -module over the ring R . The axioms oF a right R -module are ( x + y ) r = xr + yr ; x ( r + s xr + xs xs ) r = x ( sr ) ,x 1= x. “Module” will always mean leFt module unless stated otherwise. Most oF the time, there is no reason to switch the scalars From one side to the other (especially iF the underlying ring is commutative). But there are cases where we must be very careFul to distinguish between leFt and right modules (see Example 6 oF (4.1.3)). 4.1.2 Some Basic Properties o± Modules Let M be an R -module. The technique given For rings in (2.1.1) can be applied to establish the Following results, which hold For any x M and r R . We distinguish the zero vector 0 M From the zero scalar 0 R . (1) r 0 M =0 M [ r 0 M = r (0 M +0 M r 0 M + r 0 M ] (2) 0 R x M [0 R x =(0 R R ) x R x R x ] 1
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2 CHAPTER 4. MODULE FUNDAMENTALS (3) ( r ) x = r ( x )= ( rx ) [as in (2) of (2.1.1) with a replaced by r and b by x ] (4) If R is a Feld, or more generally a division ring, then =0 M implies that either r R or x M . [If r 6 = 0, multiply the equation M r 1 .] 4.1.3 Examples 1. If M is a vector space over the Feld R , then M is an R -module. 2. Any ring R is a module over itself. Rather than check all the formal requirements, think intuitively: Elements of a ring can be added and subtracted, and we can certainly multiply r R x R , and the usual rules of arithmetic apply. 3. If R is any ring, then R n , the set of all n -tuples with components in R ,i san R -module, with the usual deFnitions of addition and scalar multiplication (as in Euclidean space, e.g., r ( x 1 ,...,x n )=( 1 ,...,rx n ), etc). 4. Let M = M mn ( R ) be the set of all m × n matrices with entries in R . Then M is an R -module, where addition is ordinary matrix addition, and multiplication of the scalar c by the matrix A means multiplication of each entry of A c . 5. Every abelian group A is a Z -module. Addition and subtraction is carried out according to the group structure of A ; the key point is that we can multiply x A by the integer n .I f n> 0, then nx = x + x + ··· + x ( n times); if n< 0, then nx = x x −···− x ( | n | times).
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Chapter4 - Chapter 4 Module Fundamentals 4.1 Modules and...

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