Two (different ?) definitions of a Gröbner-Basis

Question

I have two slightly different definitions for Gröbner-Bases.
1.Definition from book
Let $I$ be an ideal and $G=(g_1,ldots,g_s)$ a basis for $I$. $G$ is called a Gröbner-Basis if $langle LT(g_1),ldots,LT(g_s) rangle = langle LT(I) rangle$ where $LT(I) = {cx^{alpha}:; text{there exists}; fin I;colon; LT(f)=cx^{alpha}}$
2.Definition in lecture
Let $I$ be an ideal and $G=(g_1,ldots,g_s)$ a basis for $I$. $G$ is called a Gröbner-Basis if $langle LM(g_1),ldots,LM(g_s) rangle = LM(I)$ where $LM(I)={LM(f);colon;0neq f in I}$
LM=Leading Monomial, LT=Leading Term
I am not so much confused that in my lecture $LM$ is used instead of $LT$, but rather that in my lecture there is "just" $LM(I)$ and not the ideal generated by $LM(I)$. Why is that ?

Ricardo Buring · Accepted Answer

The set $mathbb{M}$ of all monomials forms a semigroup, and the set of leading monomials $LM(I)$ is a semigroup ideal in $mathbb{M}$. The ideal basis $G=(g_1,ldots,g_s)$ is a Gröbner basis of $I=langle g_1,ldots,g_srangle$ iff $LM(I)$ is generated as a semigroup ideal by $LM(g_1), ldots, LM(g_s)$.
The two definitions are equivalent.

Two (different ?) definitions of a Gröbner-Basis

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