It should not be confused with a quotient ring.

In this context, the quotient ring concept is one of the most powerful unifiers.

It is the same construction used for quotient groups and quotient rings.

In every quotient ring, the nilradical is equal to the Jacobson radical.

Generalizing the previous example, quotient rings are often used to construct field extensions.

An easy motivational example is the quotient ring for any integer .

If is a domain, then and the total quotient ring is the same as the field of fractions.

For quotient rings R/I, the change of rings is also very clear.

Every integral quotient ring of R has vanishing Jacobson radical.

Let be the quotient ring of T by the ideal generated by elements .

The quotient ring R/I is a simple ring.

Let be a monic polynomial of degree , and consider the quotient ring .

Hence we may localize the ring at the set to obtain the total quotient ring .

It is important to be able to consider modules over subrings or quotient rings, especially for instance polynomial rings.

Ideals and quotient rings can be defined for rngs in the same manner as for rings.

Alternatively, note that the quotient ring has zero divisors so it is not an integral domain and thus P + Q cannot be prime.

This fits with the general rule of thumb that the smaller the ideal I, the larger the quotient ring R/I.

Since in the construction contains no zero divisors, the natural map is injective, so the total quotient ring is an extension of .

The last fact implies that actually any surjective ring homomorphism satisfies the universal property since the image of such a map is a quotient ring.

A quotient ring A/R is reduced if and only if R is a radical ideal.

Then the quotient ring R/I is the ring of germs of C-functions on M at p.

The quotient ring of a ring, is analogous to the notion of a quotient group of a group.

A number of similar statements relate properties of the ideal I to properties of the quotient ring R/I.

The starting point is a Noetherian, regular, n-dimensional ring and a full flag of prime ideals such that their corresponding quotient ring is regular.

The first isomorphism theorem for rings states that this quotient ring is naturally isomorphic to the image of f (which is a subring of S).