MATH535_HW11 - Serge Ballif MATH 535 Homework 11 1-2 Let I...

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Serge Ballif MATH 535 Homework 11 November 30, 2007 1-2) Let I be an ideal of the ring R , and let M, N be R -modules such that IN = 0 . Then show that, as R/I modules, M R N M/IM R/I N with the isomorphism “given by” m R n 7→ [ m ] R/I n . Remark: Be sure to base your argument on the Universal Mapping Property of tensor products. For example, you may induce maps between the two sides via their universal mapping properties and show that the isomorphism satisfies the property stated. Or you may simply show that one object satisfies the Universal Mapping Property of the other. (Recall the strict warning against trying to define maps from M R N by what they do on elements m R n .) Although nothing much exciting is going on, it is beneficial to work through such an example to gain experience in dealing with tensor products. Claim 1. M R N is an R/I -module. Proof. Scalar multiplication is this case is given by ( r + I )( m n ) = rm n = m rn . This map is well defined because for r + i = r 0 we have [ r 0 ]( m n ) = ( r + i )( m n ) = ( rm + im ) n = rm n + im n = rm n + m in = rm n = [ r ]( m n ) . M R N satisfies the properties of being an R/I module as a consequence of the fact that it is an R -module.
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