Discussion
Which of the following rings need not have identity ?
Answer: D
The ring that need not have an identity is the ring of polynomials \(P_{0}[\mathbf{R}]\) vanishing at the origin.
The polynomials with zero constant term are closed under addition and multiplication, so they form a ring. Suppose \(e(x)\) were an identity. Then \(e(x)\cdot x = x\) for the polynomial \(x\), so \(e(x) = 1\), and the constant 1 does not vanish at the origin. So no identity exists in this ring.
The other rings all have one. Homomorphisms of a vector space to itself (linear maps) form a ring whose identity is the identity map. Upper triangular \(n \times n\) matrices have the identity matrix \(I\), which is upper triangular. The automorphisms contain the identity map too; as printed this option is loosely worded, because invertible maps are not closed under addition and so are not a ring in the strict sense, but whichever way it is read, the identity map belongs to it.
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