Quotient Ring Defined by Ring Itself is Null Ring
Jump to navigation
Jump to search
Theorem
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.
Let $\struct {R / R, +, \circ}$ be the quotient ring defined by $R$.
Then $\struct {R / R, +, \circ}$ is a null ring.
Proof
From Ring is Ideal of Itself, it is clear we can form the quotient ring $\struct {R / R, +, \circ}$.
But $R / R = 0_R$ and so is the null ring.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 22$. Quotient Rings: Example $40$