Guided Example #1 - Algebraic Structures

The following question is of the type that may be asked on the CSET exam. In contrast to our other guided examples, check each of the four answers, so you can better understand why some answers are false and one is correct.

Which of the following properties are necessary in order to establish that the set $ℤ$ $\mathbb{Z}$ $+ i = {a + b i | a, b \in$ $ℤ$ $\mathbb{Z}$ }, together with the usual operations of addition and multiplication, is a ring? [Remark: do not confuse necessary with sufficient; there may be other properties that are also necessary to prove for the set to be a ring.]

For any element in the set of integers, $ℤ$ $\mathbb{Z}$ :

I. $(a + b i) + (c + d i) = (a + c) + (b + d) i$ , which is again an element of the set $\mathbb{Z}$ $\mathbb{Z}$ $\mathbb{Z}$ $\mathbb{Z}$ +i.

II. $(a + b i) + (c + d i) = (a c - b d) + (a d + b c) i$ , which is again an element of the set $\mathbb{Z}$ $\mathbb{Z}$ $\mathbb{Z}$ $\mathbb{Z}$ +i.

III. For every $(a + b i)$ in the set, there is the conjugate element ( $a - b i)$ , also in the set.

IV. The identity elements for addition and multiplication, 0 and 1, are both in the set.

V. $(a + b i) (c + d i) = (c + d i) (a + b i)$ since $(a c - b d) + (a d + b c) i = (c a - d b) + (d a + c b) i$

False. A ring does not have to have a multiplication that is commutative, nor does it have to have the multiplicative identity as an element. A field, however, would require both of those properties.