Question

(max new score 9/12) For this problem use the ring Z₄⊕Z₅

. This is a ring of ordered pairs.
(a) How many elements does the ring contain?
(b) In the table below on the left, list every unit of the ring, along with its inverse.
(c) In the table below on the right, list every zero divisor of the ring, along with an equation showing that it is a zero divisor.

Answer 1

In the ring Z₄⊕Z₅, there are 20 elements, including units and zero divisors. The units in the ring have inverses such that their product is the additive identity. The zero divisors are elements that, when multiplied with another element, result in the additive identity.

Step-by-step explanation:

(a)

In the ring Z₄⊕Z₅, each element is an ordered pair. The first element can be any element from Z₄, and the second element can be any element from Z₅. Since Z₄ has 4 elements and Z₅ has 5 elements, the total number of elements in the ring Z₄⊕Z₅ is 4 * 5 = 20 elements.

(b)

A unit in the ring has an inverse, such that their product is the additive identity, which is (0, 0) in this case. To find the units, we need to find elements that have inverses. The units in the ring Z₄⊕Z₅ are:

• (1, 1) with inverse (3, 4)
• (1, 2) with inverse (3, 3)
• (1, 3) with inverse (3, 2)
• (1, 4) with inverse (3, 1)
• (2, 1) with inverse (2, 4)
• (2, 2) with inverse (2, 3)
• (2, 3) with inverse (2, 2)
• (2, 4) with inverse (2, 1)
• (3, 1) with inverse (1, 4)
• (3, 2) with inverse (1, 3)
• (3, 3) with inverse (1, 2)
• (3, 4) with inverse (1, 1)

(c)

A zero divisor is an element that when multiplied with another element results in the additive identity. In the ring Z₄⊕Z₅, the zero divisors and their equations are:

• (0, 1) with equation (0, 0) = (0, 1) * (0, 1)
• (0, 2) with equation (0, 0) = (0, 2) * (0, 2)
• (0, 3) with equation (0, 0) = (0, 3) * (0, 3)
• (0, 4) with equation (0, 0) = (0, 4) * (0, 4)