The discussion revolves around the statement regarding rings with unity and the existence of units within such rings. The original poster questions whether every ring with unity has at least two units, providing reasoning based on the properties of unity and negative elements.
The discussion is ongoing, with participants examining the validity of the original claim and exploring various examples of rings. Some guidance has been offered regarding the nature of units and the characteristics of specific rings.
Participants note the zero ring as an example of a ring with unity that does not contain any units, raising questions about the definitions and properties required to discuss units in rings.
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?
I can only think of the zero ring as an example. Are there non-trivial examples?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 ?
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.Mr Davis 97 said:I can only think of the zero ring as an example. Are there non-trivial examples?