Is the mentioned basis a Gröbner basis?

Question

It's mentioned into my notes that if the ideal given as $I=\langle x+y+z, 3x-2y\rangle$, then $\{x+y+z, 5y+3z\}$ is a Gröbner basis for the ideal. I can see how $I=\langle x+y+z, 3x-2y\rangle=\langle x+y+z, 5y+3z\rangle$. The initial of the ideal with respect to the lexicographic ordering $x>y>z$, denoted as, $in_<(I)=\langle in_<(x+y+z),in_<(3x-2y)\rangle=\langle x,3x\rangle=\langle x\rangle$. By the notes definition of a Gröbner basis, $in_<(I) =\langle in_<(x+y+z), in_<(5y+3z)\rangle=\langle x,5y\rangle$, but clearly $\langle x\rangle\neq\langle x,5y\rangle$. Is the example in my notes wrong, or am I getting the definition of a Gröbner basis wrong. If it is wrong, can someone please provide a correct basis.

Answer 1

If $I=(f_1,\dots,f_m)$, then we don't necessarily have $\operatorname{in}_{<}(I)=(\operatorname{in}_{<}(f_1),\dots,\operatorname{in}_{<}(f_m))$. (Recall that by definition $\operatorname{in}_{<}(I)$ is generated by $\operatorname{in}_{<}(f)$ for all $f\in I$.) The equality holds if and only if $f_1,\dots,f_m$ is a Gröbner basis of $I$.