**Definition 1**. Let be a two-sided ideal of a ring We say that an idempotent element can be **lifted **if for some idempotent element

**Definition 2**. A subset of a ring is called **nil** if every element of is nilpotent.

**Problem**. If is a nil ideal of then every idempotent of can be lifted.

**Solution**. Suppose that is an idempotent element of Then and thus for some integer Therefore

So if we let

then and Now let Using the fact that we will get Also, since we have for all Therefore

Thus

**Example**. Let be a (left) Artinian ring and let be the Jacobson radical of Since is nilpotent, and hence nil, every idempotent of can be lifted.

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The power of $r$ in the first summation should be $i-1$.

But if $R$ has no multiplicative identity,

what is $r^0$ means?

Here is another solution,

http://goo.gl/zEK01L

http://goo.gl/FpncJD

Yes, you’re right. It should be . I just fixed it, Thank you!

In this blog all rings have unless otherwise specified. Thanks for the link though! 🙂