Fulton 2.7

Posted on November 15, 2017

2.38.* Show that if k\subset R_i, and each R_i is finite dimensional over k, then \dim(\prod R_i)=\sum\dim R_i.

Let R,S be rings, \{r_i\},\{s_j\} finite basis for R,S over k. Then \{(r_i,0)\}\cup\{(0,s_j)\} is clearly a linearly independent spanning set for R\times S. Thus \dim(R\times S)=\dim R+\dim S. The result then follows by induction.

2.39.* Prove the following relations among ideals I_i, J in a ring R:

(a) (I_1+I_2)J=I_1J+I_2J.

Suppose x=(i_1+i_2)j\in (I_1+I_2)J. Then x=i_1j+i_2j\in I_1J+I_2J. Suppose x=i_1j_1+i_2j_2\in I_1J+I_2J. Then i_1j_1,i_2j_2\in (I_1+I_2)J and thus x\in (I_1+I_2)J. So (I_1+I_2)J=I_1J+I_2J.

(b) (I_1\ldots I_N)^n=I_1^n\ldots I_N^n.

Observe that (a_1\ldots a_N)^n=a_1^n\ldots a_N^n where a_k\in I_k. Thus (I_1\ldots I_N)^n=I_1^n\ldots I_N^n.