I will be direct. This is an assignment question and I am not looking for someone to solve it for me. I want to do it myself and I am only looking for some direction and resources I can use.
- Find the basis of $$R_f=\mathbb{Z}+f\mathbb{Z}^2 \subset \mathbb{Z}^2, f\in \mathbb{Z}$$ and $\mathbb{Z}$ is the subring $(1,1)\mathbb{Z}$ in $\mathbb{Z}^2$.
What I know : $$R_f=(1,1)\mathbb{Z}+f((1,0)\mathbb{Z}+(0,1)\mathbb{Z})$$ and that $(1,1)(f,0)(0,f)$ may not be the basis over $\mathbb{Z}^2$. I went about doing the standard elimination which one does for finding basis in $\mathbb{R}^n$ but that is not helping me. Also, searching a bit tells me that it the procedure is not the same over $\mathbb{Z}^n$ as in $\mathbb{R^n}$ so how do I do it? Any online resources I can use or please somebody help me how it is done?
- Also, how does one find a basis in any ring or factor ring like $\mathbb{Z}[x]/(x^2)$?
Thanks in advance.
I'm assuming you are considering $R_f$ as a module over $\mathbb Z$ (there are no vector spaces here, as we don't have any fields hanging around).
So $$ R_f=\{(z,z)+(nf,mf):\ z,n,m\in\mathbb Z\}=\{z(1,1)+n(f,0)+m(0,f):\ z,n,m\in\mathbb Z\} $$
The elements $(1,1),(f,0),(0,f)$ are not linearly independent: $$-f(1,1)+(f,0)+(0,f)=(0,0).$$ We cannot have $(1,1)$ as a linear combination of $(f,0)$ and $0,f)$ (if $f\ne1$). But $$ (0,f)=f(1,1)-(f,0), $$ for instace. So $(1,1),(f,0)$ is a basis for $R_f$, and so is $(1,1),(0,f)$.
For something like $\mathbb Z[x]/(x^2)$, you have that $p+(x^2)=q+(x)^2$ if and only if $p-q\in (x^2)$. So, if $p=a+bx+r(x)$, $q=c+dx+s(x)$ with $r(x),s(x)\in(x^2)$, then $p-q\in(x^2)$ means that $a=c$, $b=d$. So a basis for $\mathbb Z[x]/(x^2)$ (over $\mathbb Z$) is given by $$ 1+(x^2),\ x+(x^2). $$