Ring Theory
Lecture 1A: Definition of Ring; Examples of Ring including polynomial ring.
Lecture 1B: Examples of ring; The ring of real Quaternions is a division ring; Ring of real Quaternions is an example of a division ring which is not a field.
Lecture 1C: Examples of ring.
Lecture 2A: Properties of rings; Binomial Theorem in a Commutative ring .
Lecture 3A: Definitions and examples of left zero divisor, right zero divisor zero divisor.
Lecture 3B: Definition of Integral domain; A ring (R,+, .) has no non-zero left zero divisor if and only if (R,+, .) has no non-zero right zero divisor if and only if ab=0 implies either a=0 or b=0 if and only if a is not equal to zero and b is not equal to zero implies ab is not equal zero if and only if (R-{0}, .) is a semigroup if and only if (R-{0}, .) is a semigroup in which cancellation laws hold if and only if In (R,+, .) restricted cancellation laws hold.
Lecture 3C: Finite intergral domain is division ring/skew field; Finite integral domain is field; C[0,1] is not integral domain; Ring of integral matrices is not integral domain; Z_m is integral domain if and only if m is prime.
Lecture 3D: The additive order of each non-zero element of an integral domain is same; Characteristic of Integral Domain with examples; Characteristic of Ring with identity with examples; Characteristic of integral domain is either a prime number or zero; Order of finite integral domain (that is finite field) is a power of prime; Example of an infinite integral domain whose chracteristic is finite.
Lecture 4A: Subring definition and test; Examples of ring with identity and a subring without identity; Example of a ring with identity whose subring has different identity; Example of a ring without identity whose subring has identity.
Lecture 4B: Definition and examples of left ideal, right ideal, ideal (two sided ideal).
Lecture 4C: Definition of sum of two ideals and product of two ideals with examples.
Lecture 4D: Intersection of non-empty family of subrings is a subring; Intersection of non-empty family of ideals is an ideal; The concept of subring generated by subset; The concept of ideal generated by subset; Examples and definition of principal ideal in a ring with identity.
Lecture 5A: Quotient ring with example.
Lecture 5B: Example of quotient ring; Condition under which quotient of a ring is integral domain; Prime ideal with examples.