Math Amateur
Gold Member
MHB
- 3,920
- 48
In D. G. Northcott's book: Lessons on Rings, Modules and Multiplicities, Proposition 1 reads as follows:View attachment 3453The first line of the above proof reads as follows:
"Since $$0_R + 0_R = 0_R$$, the definition of an R-module shows that
$$0_Rx = (0_R + 0_R)x = 0_Rx + 0_Rx,$$
whence $$0_Rx = 0_M$$, because $$M$$ is a group. ... ... "Now it seems highly plausible that
$$0_Rx = 0_Rx + 0_Rx$$
in the group $$M$$ leads to the conclusion that $$0_Rx = 0_M$$ ... ... BUT ... how do we know (prove) this? ...
Can someone help?
[NOTE:
Northcott seems to be saying that in a group M, for an element $$a$$:
$$a = a + a$$
$$\Longrightarrow a$$ is the identity
$$\Longrightarrow x + a = a + x = x$$ for all $$x \in M$$
... BUT ... how do we prove this? ]
"Since $$0_R + 0_R = 0_R$$, the definition of an R-module shows that
$$0_Rx = (0_R + 0_R)x = 0_Rx + 0_Rx,$$
whence $$0_Rx = 0_M$$, because $$M$$ is a group. ... ... "Now it seems highly plausible that
$$0_Rx = 0_Rx + 0_Rx$$
in the group $$M$$ leads to the conclusion that $$0_Rx = 0_M$$ ... ... BUT ... how do we know (prove) this? ...
Can someone help?
[NOTE:
Northcott seems to be saying that in a group M, for an element $$a$$:
$$a = a + a$$
$$\Longrightarrow a$$ is the identity
$$\Longrightarrow x + a = a + x = x$$ for all $$x \in M$$
... BUT ... how do we prove this? ]
Last edited: