A simplified definition and examples of prime radical of a non-commutative ring

Question

please provide a definition and some examples of prime radical (or Baer-McCoy radical or lower nilradical) of a non-commutative ring.please be specific.

Answer 1

The prime radical of a ring is defined to be the intersection of all prime ideals in the ring.

It is the smallest semiprime ideal of the ring. A ring is semiprime iff it has a zero prime radical.

This is equivalent to the nilradical for commutative rings, but this is not the case for all rings. For example, in a matrix ring $M_n(F)$ over a field $F$ with $n>1$, there are nonzero nilpotent elements, and yet the prime radical is zero. However, if one defines strongly nilpotent elements, then you can show that the prime radical is the largest ideal whose elements are all strongly nilpotent.

The prime radical is always contained in the Jacobson radical since the Jacobson radical is the intersection of right primitive ideals, all of which are prime ideals. Of course they do not have to be equal. We already know a nonfield local domain provides an example (commutative or not), but here is another example which has a nonzero prime radical:

Take the upper triangular matrices $T_2(D)$ over a nonfield local domain $D$. Let $M$ be the unique maximal right ideal of $D$. Then the prime radical of $T_2(D)$ is the set $\begin{bmatrix}0&D\\0&0\end{bmatrix}$, but the Jacobson radical is the set $\begin{bmatrix}M&D\\0&M\end{bmatrix}$.

You may notice, though, that (several formulations of) the Artin-Wedderburn theorem demonstrate that the prime radical and Jacobson radical coincide for right (or left) Artinian rings.