Show that a subset S of a ring R forms a sub ring of R if on

Show that a subset S of a ring R forms a sub ring of R if only if it contain the additive identity 0 of R and is closed under substraction and product. Use this criterion to verify that the following sets form rings when endowed with their usual operations of addition s of addition and multiplication:

Solution

Let R be a ring and S a subset of R

Let S contain 0 and closed under subtraction and product

Take any two elements in x and y in S

Since closed under subtraction 0-x = inverse of x in in S for all x in
S

x+y is in S since -y is in S and x-y is in S hence closure is true under addition

Multiplication is closed (given)

Associative since a subset of a ring

Distributive since a subset of a ring

Thus forms a subring.

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b) (i) nZ is a subset of Z

closed under multiplication and subtraction.

0 belongs to nz for all n

Hence a subring.

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(ii) 2zxz will be of the form (2n, n)

Identity elements belongs to this subset

when n =0

closed under multiplication and subtraction.

Hence a subring.