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A question about submodules...by Artusartos
Tags: submodules 
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#1
Jan1413, 05:05 PM

P: 247

My textbook says that...
If M is a left Rmodule, then a submodule N of M...is an additive subgoup N of M closed under scalar multiplication: [tex]rn \in N[/tex] whenever [tex]n \in N[/tex] and [tex]r \in R[/tex]. So if we want to prove that something is a submodule, we need to show that... 1) It closed under scalar multiplication 2) The additive idenitity is in N 3) N is closed under additition 4) If x is in N, then so is its inverse Right? But, in the link that I attached, it only shows 1) and 3), right? Can anybody tell me why? Is the proof still considered complete? Thanks in advance 


#2
Jan1413, 05:57 PM

Sci Advisor
HW Helper
P: 4,300

Isn't it as simple as: If (1) holds, set r = 0 to get (2) and r = 1 to get (4)?
(OK, you might want to show that 0n is the additive identity for any n and that n is the additive inverse of any n). 


#3
Jan1413, 06:05 PM

P: 247




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