Question

Prove or disprove: The set of units in a ring$R$ with identity is a subring of$R$ .

Short answer

Hence it is disproved that the given set of units in a ring $R$ with identity is a subring of $R$.

Step-by-step solution

Elements of Rings:

If a ringhas an element$k$ , such that$k\in R$ .

Then, the following equation will have a unique solutionas:

$k+x=0_R$

Proof:

The given ring is$R$

Let there be a set $\mathrm{\mathbb{Z}}$ which is a ring, such that the units in $\mathrm{\mathbb{Z}}$ are:

$-1and1$.

Now, if$\mathrm{\mathbb{Z}}$ is a ring, then

$1-\left(-1\right)=2\notin \left\{-1,1\right\}$.

The closure property is not satisfied, as this cannot be the subring of$R$.

Hence it is disproved that the given set of units in a ring$R$ with identity is a subring of$R$ .