0

I am newbite in abstract algebra and learning it from books by myself. I saw the following definition in my book:

Let R be a commutative ring with unity.Then, R is called a field if every nonzero element of R is a unit.

My question is that what the nonzero element of R means. Is it the number 0 ? or is it the identity element of the addition operator ? Other books also write it such as $R-{0}\$ , so I confused here. Why they use "nonzero element " term there. Can you give me more clear definition ?

If it is wanted to say the identity element of the addition operation can we say that:

Let R be a commutative ring with unity.Then, R is called a field if every element of R except for the identity element of the addition operation is unit,i.e,every element of R except for the identity element of the addition operation has inverse element with respect to multiplication operation.

Sorry for this basic question,but there is none around me to ask it, so I depend on this site as the last way.

  • 2
    Well $0$ is the usual notation for the "identity element for addition" so it is not clear what is bothering you. – Mikhail Katz Dec 24 '23 at 13:41
  • @MikhailKatz so is my definition correct ? "Let R be a commutative ring with unity.Then, R is called a field if every element of R except for the identity element of the addition operation has unit,i.e,every element of R except for the identity element of the addition operation has inverse element with respect to multiplication operation." –  Dec 24 '23 at 13:43
  • That's fine except why do you write "has unit" ? It should be "is a unit" in the sense that it has a multiplicative inverse. – Mikhail Katz Dec 24 '23 at 13:44
  • @MikhailKatz yes you are right, edit coming –  Dec 24 '23 at 13:48
  • @MikhailKatz by the way, should the identity of multiplication always different than the identity of the addition ? If they are the same , can we say that there is unity element,i.e there is the identity element of the multiplication ? –  Dec 24 '23 at 15:24
  • jerry, the simple answer is that equality 0=1 leads to a contradiction in a field because 1 must be invertible whereas 0 is not invertible. The complicated answer is that, nonetheless, there is an advanced mathematical topic of "a field with one element"; see https://mathoverflow.net/questions/tagged/field-with-one-element – Mikhail Katz Dec 24 '23 at 15:29
  • @MikhailKatz actually the question was for the ring,so when we think that field are also ring, then the identity of addititon cannot be equal to the identity of the multiplication. They are only equal in zero ring. Am i right ? –  Dec 24 '23 at 16:02

2 Answers2

0

When talking about rings and fields, zero always refers to the additive identity, and commonly 1 may be used to refer to the multiplicative identity.

Given that rings and fields don't even need to contain the standard numbers (as integers, or reals, or whatever) this is an important distinction.

ConMan
  • 27,579
  • so is my definition correct ? "Let R be a commutative ring with unity.Then, R is called a field if every element of R except for the identity element of the addition operation has unit,i.e,every element of R except for the identity element of the addition operation has inverse element with respect to multiplication operation." –  Dec 24 '23 at 13:43
  • Yes, that definition is correct. – ConMan Dec 24 '23 at 13:47
  • actually the question was for the ring,so when we think that field are also ring, then the identity of addititon cannot be equal to the identity of the multiplication. They are only equal in zero ring. Am i right ? –  Dec 24 '23 at 17:06
0

Your understanding is correct.

$0$ or "zero-element" is just a short hand for "identity of the additive abelian group $(R,+)$".

Every ring must have a zero-element as a consequence of the definition. Don't think of this $0$ as the real number zero, it's more general.

For a ring with identity, $R^*$ is used to indicate the group of units. $$R^*=\{x\in R \text{ such that } \exists x^{-1}\in R\}$$

In case $R$ is a field, $R^*= R\setminus\{0\}$.

Nothing special
  • 3,690
  • 1
  • 7
  • 27
  • actually the question was for the ring,so when we think that field are also ring, then the identity of addititon cannot be equal to the identity of the multiplication. They are only equal in zero ring. Am i right ? –  Dec 24 '23 at 17:06
  • @jerry17 Yes, generally, zero-rings are NOT considered field. You are right. In a field, it is necessary that $0\neq 1$. The only ring where $0=1$ holds is the zero-ring (which is not a field). Mind that $1$ is just a symbol for the identity of multiplication. – Nothing special Dec 24 '23 at 17:39