Register to reply

A question about submodules...

by Artusartos
Tags: submodules
Share this thread:
Artusartos
#1
Jan14-13, 05:05 PM
P: 247
My textbook says that...

If M is a left R-module, 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
Attached Thumbnails
636.jpg  
Phys.Org News Partner Science news on Phys.org
NASA team lays plans to observe new worlds
IHEP in China has ambitions for Higgs factory
Spinach could lead to alternative energy more powerful than Popeye
CompuChip
#2
Jan14-13, 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).
Artusartos
#3
Jan14-13, 06:05 PM
P: 247
Quote Quote by CompuChip View Post
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).
Thanks.

mathwonk
#4
Jan15-13, 09:33 PM
Sci Advisor
HW Helper
mathwonk's Avatar
P: 9,453
A question about submodules...

1 does not imply 2, unless the subset considered is non empty. i.e. 1 implies that IF the subset contains anything, then it also contains 0.


Register to reply

Related Discussions
Radical ideals and submodules Linear & Abstract Algebra 0
Nonzero R-Module over commuttaive ring, all submodules free => R PID? Linear & Abstract Algebra 6
Irreducible Modules/Submodules & Group Algebras (G = D6) Calculus & Beyond Homework 0
Problems regarding group presentations and submodules Linear & Abstract Algebra 1