Question

prove that if p is an element of a commutative ring R, then p ={p.r:r∈p} is an ideal in R.

Answer 1

Can you clarify the set p a little? What I see is: $$p=\{p\cdot r\mid r\in p\},$$which is a little circular for a definition (the set p is in how you are defining it). Do you mean: $$(p)=\{p\cdot r\mid r\in R\}?$$

Answer 2

am not also sure but suppose that is the case i.e (p)={p.r:r∈R} how do we prove it?