L#3 | 2022 | Ring Theory and Linear Algebra-I | Subrings | B.Sc. Mathematics | DU

In our study of groups, subgroups played a crucial role. Subrings, the
analogous structures in ring theory, play a much less prominent role than
their counterparts in group theory. Nevertheless, subrings are important.

Definition Subring
A subset S of a ring R is a subring of R if S is itself a ring with the
operations of R

A nonempty subset S of a ring R is a subring if S is closed under
subtraction and multiplication—that is, if a- b and ab are in S
whenever a and b are in S.
EXAMPLE 12 Let R be the ring of all real-valued functions of a single real variable under pointwise addition and multiplication. The subset S of R of functions whose graphs pass through the origin forms a subring of R.

1. Give an example of a finite noncommutative ring. Give an example
of an infinite noncommutative ring that does not have a unity.
2. The ring {0, 2, 4, 6, 8} under addition and multiplication modulo
10 has a unity. Find it.
3. Give an example of a subset of a ring that is a subgroup under
addition but not a subring.
4. Show, by example, that for fixed nonzero elements a and b in a
ring, the equation ax 5 b can have more than one solution. How
does this compare with groups?

[yyttpop]

Sir when will you start lectures on linear algebra - I ?Please tell

[deepashachhabramath]

Sir, if you can have one lecture more on questions it
Will be helpful.