Abstract:
Radical theory in rings is regarded as a major theoretical concept that specifies
certain classical structure theorems for rings, enhancing profound understanding
of the key aspects of ring theory. Radical theory has aided to the development of
mathematics by providing a background to study and compare properties using
radical-like theories, constructing rings with distinguished properties, and opening
new dimensions for research. The research focused on gathering theories on
radicals in rings purely from past research, filtering and combining them, that
would help a reader by providing a brief yet clear understanding on the key
concepts of radical theories. The research objectives are of four-fold: with respect
to radicals in ring theory (i) to study general theories of radicals, (ii) to identify
classical examples of radicals and their properties, (iii) to critically apply general
theories of radicals to examples, (iv) and to find out the usefulness by generalizing
the radical concept to abstract algebraic systems. The research method included
collection of resources (electronic books and articles) from online portals, a
methodical skimming of the resources, classification of the main topic into relevant
subtopics, writing and compiling the grabbed knowledge into own appropriate
words, analysis of the connection between examples with valid reasonings, and
making of a final review presentation by assembling all the gathered information
into appropriate components. In general, radical (P(R)) of a ring (R) is an ideal of
“bad” elements of the ring (R) with an inappropriate or bad property (P). Inventive
technique is used to classify bad elements in a ring and find radicals to make them
good, reducing the radical of a quotient ring (R/P(R)) to zero ideal; P(R/P(R)) =
(0). Key examples of radicals are Jacobson radical, Nilradical, Levitzki radical,
Artinian radical, Brown-McCoy radical, Baer radical, etc. The bad property of
these examples is generally related to zero multiplication or nilpotency. For a ring
R, different kinds of radicals can be related by inclusions; B(R) ⊆ L(R) ⊆ J(R) ⊆G(R), where B denotes the Baer radical, L denotes the Levitzki radical, J denotes
the Jacobson radical, and G denotes the Brown-McCoy radical. Theories were
developed using the concepts of minimal right ideals, nilpotency and radical liketheories. With the modern research, mathematicians are generalizing the radical
concepts to other algebraic systems like Banach algebra, Group algebra, Near
rings, Lattices, etc.
