Radical of a ring

In ring theory, a branch of mathematics, a radical of a ring is an ideal of "not-good"^{[definition needed]} elements of the ring.

The first example of a radical was the nilradical introduced by Köthe (1930), based on a suggestion of Wedderburn (1908). In the next few years several other radicals were discovered, of which the most important example is the Jacobson radical. The general theory of radicals was defined independently by (Amitsur 1952, 1954, 1954b) and Kurosh (1953).

The study of radicals is called torsion theory.

In the theory of radicals, rings are usually assumed to be associative, but need not be commutative and need not have a multiplicative identity. In particular, every ideal in a ring is also a ring.

Let ${\displaystyle {\mathfrak {A}}}$ be a class of rings which is:

In particular, ${\displaystyle {\mathfrak {A}}}$ could just be the class of all (non-unital) rings.

Let r be some abstract property of rings in ${\displaystyle {\mathfrak {A}}}$. A ring with property r is called an r-ring; an ideal of some ring with property r is called an r-ideal. In particular, the r-ideals are a subset of the r-rings. A ring ${\displaystyle A}$ is said to be a r-semi-simple ring if it has no non-zero r-ideals.

r is said to be a radical property if: