# Every ring with unity has at least two units

• Mr Davis 97
In summary, the conversation discusses the statement "Every ring with unity has at least two units" and whether it is true or false. The conversation considers the definition of units in a ring and whether the elements "a" and "-a" are necessarily distinct. It is concluded that there are many non-trivial examples of rings where 1 = -1, and therefore the statement is false.
Mr Davis 97

## Homework Statement

T/F: Every ring with unity has at least two units

## The Attempt at a Solution

I thought that the answer was true, because if a ring ##R## has unity ##1##, then ##1 \cdot 1 = 1## and ##(-1) \cdot (-1) = 1##. Where am I going wrong?

Mr Davis 97 said:
I thought that the answer was true, because if a ring ##R## has unity ##1##, then ##1 \cdot 1 = 1## and ##(-1) \cdot (-1) = 1##. Where am I going wrong?

The elements denoted by "##a##" and "##-a##" are not necessarily distinct from each other. Can you think of a ring where 1 = -1 ?

Stephen Tashi said:
The elements denoted by "##a##" and "##-a##" are not necessarily distinct from each other. Can you think of a ring where 1 = -1 ?
I can only think of the zero ring as an example. Are there non-trivial examples?

Mr Davis 97 said:
I can only think of the zero ring as an example. Are there non-trivial examples?
Many. The zero ring hasn't any units at all. So what do you need also to speak of units and what is the smallest example? Based on this, there are plenty of examples.

## 1. What is a ring with unity?

A ring with unity is a mathematical structure consisting of a set of elements and two binary operations, usually addition and multiplication. The unity element, denoted by 1, is an element that acts as an identity for the multiplication operation.

## 2. What does it mean for a ring to have at least two units?

A unit in a ring is an element that has a multiplicative inverse, meaning that when multiplied together, they equal the unity element. A ring with at least two units means that there are at least two elements in the ring that have multiplicative inverses.

## 3. Why is it important for a ring to have at least two units?

Having at least two units in a ring enables the existence of a non-trivial subring, which is a subset of the original ring that can be used to study properties of the larger ring. It also allows for the definition of a multiplicative group, which has important applications in abstract algebra and number theory.

## 4. Is every ring with unity guaranteed to have at least two units?

Yes, every ring with unity is guaranteed to have at least two units. This is a fundamental property of rings and can be proven using the definition of a ring and the existence of a multiplicative inverse for the unity element.

## 5. Can you provide an example of a ring with unity that has more than two units?

Yes, the ring of Gaussian integers, which consists of all complex numbers of the form a + bi, where a and b are integers, and i is the imaginary unit, has infinitely many units. This is because every non-zero element in this ring has a multiplicative inverse.

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