Final answer:
a) R is a subring of Q because it is closed under addition, subtraction, and multiplication and contains the additive and multiplicative identity elements. b) R × is equal to R as they have the same elements. c) The ideal generated by 43 in R is n ∈ N, z ∈ Z. d) R is a principal ideal domain, as every ideal in R is principal.
Step-by-step explanation:
a) To show that R is a subring of Q, we need to prove that it is closed under addition, subtraction, and multiplication and that it contains the additive identity element (0) and the multiplicative identity element
(1). Let's consider two elements x = 2n/z and y = 2m/w in R, where n and m are natural numbers and z and w are integers. The sum of x and y is (2n/z) + (2m/w) = (2nz + 2mw) / (zw),
which is still in R since 2nz + 2mw is a multiple of 2 and zw is an integer. The product of x and y is (2n/z)(2m/w) = (4nm)/(zw), which is also in R since 4nm is a multiple of 2 and zw is an integer.
Thus, R is closed under addition and multiplication. R contains the additive identity 0 = 2*0/1 and the multiplicative identity 1 = 2*1/1. Therefore, R is a subring of Q.
b) The set R × is the set of all elements (2n/z) × (2m/w) in R, where n and m are natural numbers and z and w are integers.
The product of two elements x = (2n/z) and y = (2m/w) in R × is given by x × y = (2n/z) × (2m/w) = (4nm)/(zw),
which is still in R × since 4nm is a multiple of 2 and zw is an integer.
Therefore, R × = R.
c) The ideal generated by 43 in R is the set of all elements of the form 43 × (2n/z), where n is a natural number and z is an integer.
This can be written as n ∈ N, z ∈ Z.
The ideal generated by 43 is a subset of R, since any element of the form 86n/z is also in R. Therefore, the ideal generated by 43 in R is n ∈ N, z ∈ Z.
d) To show that R is a principal ideal domain, we need to prove that every ideal in R is principal. Let me be an ideal in R. If I is the zero ideal {0}, then it is principal, and the statement holds.
Otherwise, let x be a nonzero element in I. Since R is a subring of Q, we can write x as x = 2n/z, where n is a natural number and z is an integer. Then, the ideal generated by x in R is 2n/z × (2m/w) .
This ideal can also be written as (2nm)/(zw) , which is equal to R since 2nm is a multiple of 2 and zw is an integer. Therefore, every ideal in R is principal, and R is a principal ideal domain.