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let-r-be-a-commutative-ring-let-i-and-j-be-ideals-of-r-let-i-j-x-y-xini-yinj-and-ij-xy-xini-yi-pa749

Let R be a commutative ring. Let

`I`

and

`J`

be ideals of

`R`

. Let

`I-J={x-y|xinI,yinJ}`

and

`IJ={xy|xinI,yinJ}`

. Then (a.)

`I-J`

is an ideal and

`I`

is an ideal in

`R`

(b.)

`I-J`

is an ideal and IJ need not be an ideal in R (c.)

`I-J`

need not be an ideal but IJ is an ideal in R (d.) Neither

`I-J`

nor

`IJ`

need to be an ideal in R .