Can we say "commutative ring = field"?

Question

We know the difference between ring ($R$) and field ($F$) is that $R$ does not guarantee multiplication is commutative.

Now, if considering commutative $R$, which means ($R$, $*$) is a group, can we say: Commutative ring "$=$" field?

Thanks!

Answer 1

No, you can not say that. The flaw in your reasoning is that being commutative does not make $(R,\cdot)$ a group. It does not even make $(R\setminus\{0\},\cdot)$ a group. If that was the case, then $R$ would be a field. For $(R\setminus\{0\},\cdot)$ to be a group, every nonzero element would have to have an inverse. In the integer numbers $\mathbb Z$, for instance, multiplication is commutative, but $2$ has no multiplicative inverse - after all, $\frac12\notin\mathbb Z$.

Answer 2

Definitely not. For example, the integers under the usual addition and multiplication form a commutative ring that is not a field.

The polynomial with (say) real coefficients are a commutative ring, but not a field.

The integers $0,1,2,3,4,5$, under addition and multiplication modulo $6$, are a commutative ring but not a field.

In all the above examples, there are non-zero elements that do not have a multiplicative inverse.

Answer 3

No. $\mathbb{Z}$ is a commutative ring. It is not a field. A fundamental difference is the existence of multiplicative inverses.