[代數] 代數(9)

[TaiwanBank (PTT生日快樂^^)]

Let R be a ring with 1 and I,J be two-sided ideals of R.

(a) Give an example to show that IJ = I∩J need not be true.
(b) Suppose I+J = R.  Show that IJ = I∩J.

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推 Sfly:for a, take R=Z,(2)(2)=(4)!=(2)=(2)∩(2).
→ Sfly:for b, choose i in I nad j in J such that i+j=1, then
→ Sfly:any x in I∩J, we have x=ix+xj, which is in IJ.
→ Sfly:and the contrary is trivial.