what is $\langle 2, x , y\rangle$ in $\mathbb Z[x,y]$?

Question

Can anyone tell me what is $\langle 2 , x , y\rangle $ in $\mathbb Z[x,y]$? I was reading Dummit and Foote. But I got stuck with this notation which was not defined there.

Answer 1

An element $\;\omega\in\langle 2,x,y\rangle\;$ looks like

$$\omega=2f(x,y)+xg(x,y)+yh(x,y)\;,\;\;\text{with}\;\;f,g,h\in\Bbb Z[x,y]$$

Observe that $\;\omega\;$ as above has even free coefficient, which comes from the addend $\;2f(x,y)\;$ , since the other two addends have each free coefficient equal to zero.

OTOH, if $\;p\in\Bbb Z[x,y]\;$ has even free coefficient, say $\;2k\;$ , then we can write

$$p(x,y)=2k+\sum_{k=0}^{n}a_k(x)y^k\;,\;\;a_k\in\Bbb Z[x]\;,\;\;a_0\;\text{ non-constant with free coefficient zero}\implies$$

$$p(x,y)=k\cdot2+a_0(x)+\left(\sum_{k=1}^na_k(x)\right)y^k\in\langle 2,x,y\rangle $$

Answer 2

it is the ideal generated by these elements.

In particular, given $a_1, \dots a_n \in R$, we can ask "what is the smallest ideal containing these elements?" I,e: the intersection of all ideals containing these elements.

Specializing to your case, $\langle x\rangle$ consists of all polynomials $x \cdot P(x,y)$ so all polynomials where you can factor out an $x$, and $\langle x,y\rangle$ consists of polynomials $xP(x,y)+yP(x,y)$ so all polynomials made up of just $x,y$ (without constant terms) and so on.