TITLE: What is the ideal of leading terms?
QUESTION [5 upvotes]: Fix a monomial ordering on the polynomial ring $\Bbb{k}[x_1, \dots, x_n] = R$ over a field.  What exactly is $LT(I)$ for an ideal $I$ of $R$?  How is it defined and does it form an ideal?

REPLY [3 votes]: $LT(I)$ is the ideal generated by the leading terms of all polynomials in $I$. By definition, it is an ideal. 
As an aside, note that if an ideal is generated by polynomials $f_1, \dots, f_k$, it is not necessarily the case that $LT(I)$ is generated by the leading terms of the $f_i$. If it is, then the $f_i$ form a Groebner basis of $I$.