Question

(max total new score 6/8) (a) List ALL the subrings of Z₁₈other than {0} and Z₁₈itself.

(b) State ALL the subring relationships that exist between these five rings: Z,R,C,Z[i],Q[ √2]

Answer 1

The subrings of Z₁₈ other than {0} and Z₁₈ are Z₂, Z₃, Z₆, and Z₉. In terms of subring relationships among Z, R, C, Z[i], Q[√2], each is a subring of C, with Z being the most restrictive and also a subring of the others except Q[√2].

Step-by-step explanation:

Subrings of Z₁₈

To determine the subrings of Z₁₈ other than \{0\} and Z₁₈ itself, we need to consider the divisors of 18. A subring of a ring Zn must be of the form Zd, where d is a divisor of n. For 18, the divisors are 1, 2, 3, 6, 9, and 18. As per the question, we exclude 1 and 18, so the subrings of Z₁₈ are:

• Z2
• Z3
• Z6
• Z9

Subring Relationships

The following rings Z, R, C, Z[i], Q[√2] have specific relationships:

• Z (the integers) is a subring of the rational numbers Q, the real numbers R, the complex numbers C, and the Gaussian integers Z[i].
• Q (the rational numbers) is a subring of R and C.
• R (the real numbers) is a subring of C.
• Z[i] (the Gaussian integers) is a subring of C.
• Q[√2] (the rational numbers extended by the square root of 2) is a subring of R and C.