Subring | definition & example | subring criteria| Ring Theory| abstract algebra

In this video Subring | definition & example | subring criteria| Ring Theory| abstract algebra we will discuss an important topic of chapter Rings and modules from book abstract algebra by dummit Foote. This topic is taught in university to the students of msc mathematics BS maths bsc 3rd year.
In this video we will learn what is subring and it's examples. This lecture contains subring in ring Theory abstract algebra definition and examples of subring theorem and proof of subring.
In this video we will discuss the subring criteria in Ring Theory also called necessary and sufficient condition for a subring.
In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R.
Here's an example:
A subring of a ring (R, +, ∗, 0, 1) is a subset S of R that preserves the structure of the ring, i.e. a ring (S, +, ∗, 0, 1) with S ⊆ R. Equivalently, it is both a subgroup of (R, +, 0) and a submonoid of (R, ∗, 1).Z×Z3 is not a subring of Z×Z6, because Z3 is not a subring of Z6.
Also we will learn in this video that How do you prove a subset is a subring?
In general, to show that a subset S of a ring R, is a subring of R, it is sufficient to show that (i) S is closed under addition in R (ii) S is closed under multiplication in R; (iii) 0R ∈ S; (iv) when a ∈ S, the equation a + x = 0R has a solution in S.

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