Why are $(X_1), (X_1,X_2), \ldots$ prime ideals?

Question

I was looking at the proof of the dimension of the polynomial ring $R[X_1,\ldots,X_n]$ and I had a question: Why are $(X_1), (X_1,X_2), (X_1,X_2,X_3),\ldots, (X_1,\ldots,X_n)$ prime ideals in this ring? I couldn't even prove that they are ideals, I really need help.

EDIT

My problem is I don't understand why if $f,g\in R[X_1,\ldots,X_n]$ such that $fg+I=I\implies f$ or $g\in I$, where $I=(X_1)$

Thanks in advance.

Answer 1

The parentheses around them indicate the ideal generated by those things, so they form an ideal by definition.

Secondly, you can tell they are prime by looking at the quotient ring by each ideal. You believe polynomial rings over fields are domains, right?