## A Book of Abstract Algebra, Second Edition (1982)

### Chapter 18. IDEALS AND HOMOMORPHISMS

We have already seen several examples of smaller rings contained within larger rings. For example, is a ring inside the larger ring , and itself is a ring inside the larger ring . When a ring *B* is part of a larger ring *A*, we call *B*a *subring* of *A*. The notion of subring is the precise analog for rings of the notion of subgroup for groups. Here are the relevant definitions:

Let *A* be a ring, and *B* a nonempty subset of *A*. If the sum of any two elements of *B* is again in *B*, then *B* is *closed with respect to addition*. If the negative of every element of *B* is in *B*, then *B* is *closed with respect to negatives*. Finally, if the product of any two elements of *B* is again in *B*, then *B* is *closed with respect to multiplication. B* is called a *subring* of *A* if *B* is closed with respect to addition, multiplication, and negatives. Why is *B* then called a subring of *A*? Quite elementary:

*If a nonempty subset B ⊆ A is closed with respect to addition, multiplication, and negatives, then B with the operations of A is a ring*.

This fact is easy to check: If *a*, *b*, and *c* are any three elements of *B*, then *a*, *b*, and *c* are also elements of *A* because *B* ⊆ *A*. But *A* is a ring, so

Thus, in *B* addition and multiplication are associative and the distributive law is satisfied. Now, *B* was assumed to be nonempty, so there is an element *b* ∈ *B* but *B* is closed with respect to negatives, so −*b* is also in *B*. Finally, *B* is closed with respect to addition; hence *b* + (−*b*) ∈ *B*. That is, 0 is in *B*. Thus, *B* satisfies all the requirements for being a ring.

For example, is a subring of because the sum of two rational numbers is rational, the product of two rational numbers is rational, and the negative of every rational number is rational.

By the way, if *B* is a nonempty subset of *A*, there is a more compact way of checking that *B* is a subring of *A* :

*B is a subring of A if and only if B is closed with respect to subtraction and multiplication*.

The reason is that *B* is *closed with respect to subtraction iff B is closed with respect to both addition and negatives*. This last fact is easy to check, and is given as an exercise.

Awhile back, in our study of groups, we singled out certain special subgroups called *normal subgroups*. We will now describe certain special subrings called *ideals* which are the counterpart of normal subgroups: that is, ideals are in rings as normal subgroups are in groups.

Let *A* be a ring, and *B* a nonempty subset of *A*. We will say that *B absorbs products in A* (or, simply, *B absorbs products*) if, whenever we multiply an element in *B* by an element in *A* (regardless of whether the latter is inside *B*or outside *B*), their product is always in *B*. In other words,

for all *b* ∈ *B* and *x* ∈ *A*, *xb* and *bx* are in *B*.

*A nonempty subset B of a ring A is called an ideal of A if B is closed with respect to addition and negatives, and B absorbs products in A*.

A simple example of an ideal is the set of the even integers. is an ideal of because the sum of two even integers is even, the negative of any even integer is even, and, finally, the product of an even integer by *any* integer is always even.

In a commutative ring with unity, the simplest example of an ideal is the set of all the multiples of a fixed element *a* by all the elements in the ring. In other words, the set of all the products

*xa*

as *a* remains fixed and *x* ranges over all the elements of the ring. This set is obviously an ideal because

and

*y*(*xa*) = (*yx*)*a*

This ideal is called the *principal ideal generated by a*, and is denoted by

⟨*a*⟩

As in the case of subrings, if *B* is a nonempty subset of *A*, there is a more compact way of checking that *B* is an ideal of *A :*

*B is an ideal of A if and only if B is closed with respect to subtraction and B absorbs products in A*.

We shall see presently that ideals play an important role in connection with homomorphism

Homomorphisms are almost the same for rings as for groups.

*A homomorphism from a ring A to a ring B is a function f : A → B satisfying the identities*

*f*(*x*_{l} + *x*_{2}) = *f*(*x*_{1}) + *f*(*x*_{2})

*and*

*f*(*x*_{1}*x*_{2}) = *f*(*x*_{1})*f*(*x*_{2})

There is a longer but more informative way of writing these two identities:

1. *If f*(*x*_{1}) = *y*_{1} *and f*(*x*_{2}) = *y*_{2} *then f*(*x*_{1} + *x*_{2}) = *y*_{1} + *y*_{2}.

2. *If f*(*x*_{1}) = *y*_{1} *and f*(*x*_{2}) = *y*_{2} *then f*(*x*_{1} *x*_{2}) = *y*_{1} *y*_{2}

In other words, if *f* happens to carry *x*_{l} to *y*_{1} and *x*_{2} to *y*_{2}, then, necessarily, it must carry *x*_{1} + *x*_{2} to *y*_{1} + *y*_{2} and *x*_{1}*x*_{2} to *y*_{1}*y*_{2}. Symbolically,

*If* *and* , *then necessarily*

One can easily confirm for oneself that a function *f* with this property will transform the addition and multiplication tables of its domain into the addition and multiplication tables of its range. (We may imagine infinite rings to have “nonterminating” tables.) Thus, a homomorphism from a ring *A onto* a ring *B* is a function which transforms *A* into *B*.

For example, the ring _{6} is transformed into the ring _{3} by

as we may verify by comparing their tables. The addition tables are compared on page 136, and we may do the same with their multiplication tables:

If there is a homomorphism from *A onto B*, we call *B* a *homomorphic image of A*. If *f* is a homomorphism from a ring *A* to a ring *B*, not necessarily *onto*, the range of/is a subring of *B*. (This fact is routine to verify.) Thus, the range of a ring homomorphism is always a ring. And obviously, the range of a homomorphism is always a homomorphic image of its domain.

Intuitively, if *B* is a homomorphic image of *A*, this means that certain features of *A* are faithfully preserved in *B* while others are deliberately lost. This may be illustrated by developing further an example described in __Chapter 14__. The *parity ring P* consists of two elements, *e* and *o*, with addition and multiplication given by the tables

We should think of *e* as “even” and *o* as “odd,” and the tables as describing the rules for adding and multiplying odd and even integers. For example, even + odd = odd, even *times* odd = even, and so on.

The function *f*: → *P* which carries every even integer to *e* and every odd integer to *o* is easily seen to be a homomorphism from to *P* this is made clear on page 137. Thus, *P* is a homomorphic image of . Although the ring *P*is very much smaller than the ring , and therefore few of the features of can be expected to reappear in *P*, nevertheless *one* aspect of the structure of is retained absolutely intact in *P*, namely, the structure of odd and even numbers. As we pass from to *P*, the *parity* of the integers (their being even or odd), with its arithmetic, is faithfully preserved while all else is lost. Other examples will be given in the exercises.

If *f* is a homomorphism from a ring *A* to a ring *B*, the *kernel* of *f* is the set of all the elements of *A* which are carried by *f* onto the zero element of *B*. In symbols, the kernel of *f* is the set

*K* = {*x* ∈ *A* : *f*(*x*) = 0}

It is a very important fact that the *kernel of f is an ideal of A*. (The simple verification of this fact is left as an exercise.)

If *A* and *B* are rings, an *isomorphism* from *A* to *B* is a homomorphism which is a one-to-one correspondence from *A* to *B*. In other words, it is an injective and surjective homomorphism. If there *is* an isomorphism from *A* to *B*we say that *A* is *isomorphic to B*, and this fact is expressed by writing

*A* ≅ *B*

**EXERCISES**

**A. Examples of Subrings**

Prove that each of the following is a subring of the indicated ring:

**1** {*x* + *y* : *x*, *y* ≅ } is a subring of .

**2** {*x* + 2^{1/3} *y* + 2^{2/3} *z* : *x*, *y*, *z* ∈ } is a subring of .

**3** {*x*2* ^{y}* :

*x*,

*y*∈ } is a subring of .

**# 4** Let () be the set of all the functions from to which are continuous on (−∞, ∞) and let () be the set of all the functions from to which are differentiable on (−∞, ∞). Then () and () are subrings of ().

**5** Let () be the set of all functions from to which are continuous on the interval [0,1]. Then () is a subring of (), and () is a subring of ().

**6** The subset of _{2}() consisting of all matrices of the form

is a subring of _{2}().

**B. Examples of Ideals**

**1** Identify which of the following are ideals of × , and explain: {(*n*, *n*) : *n* ∈ }; {(5*n*, 0) : *n* ∈ }; {(*n*, *m*) : *n* + *m* is even}; {(*n*, *m*) : *nm* is even}; {(2*n*, 3*m*) : *n*, *m* ∈ }.

**2** List all the ideals of _{12}.

**# 3** Explain why every subring of

*is necessarily an ideal.*

_{n}**4** Explain why the subring of __Exercise A6__ is not an ideal.

**5** Explain why () is not an ideal of ().

**6** Prove that each of the following is an ideal of ():

(*a*) The set of all *f* such that *f*(*x*) = 0 for every rational *x*.

(*b*) The set of all *f* such that *f*(0) = 0.

**7** List all the ideals of *P*_{3}. (*P*_{3} is defined in __Chapter 17__, __Exercise D__.)

**8** Give an example of a subring of *P*_{3} which is not an ideal.

**9** Give an example of a subring of _{3} × _{3} which is not an ideal.

**C. Elementary Properties of Subrings**

Prove parts 1–6:

**1** A nonempty subset *B* of a ring *A* is closed with respect to addition and negatives iff *B* is closed with respect to subtraction.

**2** Conclude from part 1 that *B* is a subring of *A* iff *B* is closed with respect to subtraction and multiplication.

**3** If *A* is a finite ring and *B* is a subring of *A*, then the order of *B* is a divisor of the order of *A*.

**# 4** If a subring

*B*of an integral domain

*A*contains 1, then

*B*is an integral domain. (

*B*is then called a

*subdomain*of

*A*.)

**# 5** Every subring containing the unity of a field is an integral domain.

**6** If a subring *B* of a field *F* is closed with respect to multiplicative inverses, then *B* is a field. (*B* is then called a *subfield* of *F*.)

**7** Find subrings of _{18} which illustrate each of the following:

(*a*) *A* is a ring with unity, B is a subring of *A*, but *B* is not a ring with unity.

(*b*) *A* and *B* are rings with unity, *B* is a subring of *A*, but the unity of *B* is not the same as the unity of *A*.

**8** Let *A* be a ring, *f* : *A* → *A* a homomorphism, and *B* = {*x* ∈ *A* : *f*(*x*) = *x*}. Prove that *B* is a subring of *A*.

**9** The *center* of a ring *A* is the set of all the elements *a* ∈ *A* such that *ax* = *xa* for every *x* ∈ *A*. Prove that the center of *A* is a subring of *A*.

**D. Elementary Properties of Ideals**

Let *A* be a ring and *J* a nonempty subset of *A*.

**1** Using __Exercise C1__, explain why *J* is an ideal of *A* iff *J* is closed with respect to subtraction and *J* absorbs products in *A*.

**2** If *A* is a ring with unity, prove that *J* is an ideal of *A* iff *J* is closed with respect to addition and *J* absorbs products in *A*.

**3** Prove that the intersection of any two ideals of *A* is an ideal of *A*.

**4** Prove that if *J* is an ideal of *A* and 1 ∈ *J*, then *J* = *A*.

**5** Prove that if *J* is an ideal of *A* and *J* contains an invertible element *a* of *A*, then *J* = *A*.

**6** Explain why a field *F* can have no nontrivial ideals (that is, no ideals except {0} and *F*).

**E. Examples of Homomorphisms**

Prove that each of the functions in parts 1–6 is a homomorphism. Then describe its kernel and its range.

**1** *ϕ* : ()→ given by *ϕ*(*f*) = *f*(0).

**2** *h* : × → given by *h*(*x*, *y*) = *x*.

**3** *h* : → _{2}() given by

**4** *h* : × → _{2}() given by

**# 5** Let

*A*be the set × with the usual addition and the following “multiplication”:

(*a*, *b*) (*c*, *d*) = (*ac*, *bc*)

Granting that *A* is a ring, let *f* : *A* → _{2}() be given by

**6** *h* : *P _{c}* →

*P*given by

_{c}*h*(

*A*) =

*A*

*D*, where

*D*is a fixed subset of

*C*.

**7** List all the homomorphisms from _{2} to _{4}; from _{3} to _{6}.

**F. Elementary Properties of Homomorphisms**

Let *A* and *B* be rings, and *f* : *A* → *B* a homomorphism. Prove each of the following:

**1** *f*(*A*) = {*f*(*x*): *x* ∈ *A*} is a subring of *B*.

**2** The kernel of *f* is an ideal of *A*.

**3** *f*(0) = 0, and for every *a* ∈ *A*, *f*(−*a*) = −*f*(*a*).

**4** *f* is injective iff its kernel is equal to {0}.

**5** If *B* is an integral domain, then either *f*(l) = 1 or *f*(l) = 0. If *f*(l) = 0, then *f*(*x*) = 0 for every *x* ∈ *A*. If *f*(1) = 1, the image of every invertible element of *A* is an invertible element of *B*.

**6** Any homomorphic image of a commutative ring is a commutative ring. Any homomorphic image of a field is a field.

**7** If the domain *A* of the homomorphism *f* is a field, and if the range of *f* has more than one element, then *f* is injective. (HINT: Use __Exercise D6__.)

**G. Examples of Isomorphisms**

**1** Let *A* be the ring of __Exercise A2__ in __Chapter 17__. Show that the function *f*(*x*) = *x* − 1 is an isomorphism from to *A* hence ≅ *A*.

**2** Let be the following subset of _{2}():

Prove that the function

is an isomorphism from to . [REMARK: You must begin by checking that *f* is a well-defined function; that is, if *a* + *b*i = *c* + *d*i, then *f*(*a* + *b*i) = *f*(*c* + *d*i). To do this, note that if *a* + *b*i = *c* + *d*i then *a* − *c* = (*d* − *b*)i; this last equation is impossible unless both sides are equal to zero, for otherwise it would assert that a given real number is equal to an imaginary number.]

**3** Prove that {(*x*, *x*) : *x* ∈ } is a subring of × , and show {(*x*, *x*) : *x* ∈ } ≅ .

**4** Show that the set of all 2 × 2 matrices of the form

is a subring of _{2}(), then prove this subring is isomorphic to .

For any integer *k*, let *k* designate the subring of which consists of all the multiples of *k*.

**5** Prove that ∉ 2 then prove that 2 ∉ 3. Finally, explain why if *k* ≠ *l*, then *k* ∉ *l*. (REMEMBER: How do you show that two rings, or groups, are *not* isomorphic?)

**H. Further Properties of Ideals**

Let *A* be a ring, and let *J* and *K* be ideals of *A*.

Prove parts 1-4. (In parts 2-4, assume *A* is a commutative ring.)

**1** If *J* *K* = {0}, then *jk* = 0 for every *j* ∈ *J* and *k* ∈ *K*.

**2** For any *a* ∈ *A*, *I _{a}* = {

*ax*+

*j*+

*k*:

*x*∈

*A*,

*j*∈

*J*,

*k*∈

*K*} is an ideal of

*A*.

**# 3** The

*radical*of

*J*is the set rad

*J*= {

*a*∈

*A*:

*a*∈

^{n}*J*for some

*n*∈ }. For any ideal

*J*, rad

*J*is an ideal of

*A*.

**4** For any *a* ∈ *A*, {*x* ∈ *A* : *ax* = 0} is an ideal (called the *annihilator* of *a*).

Furthermore, {*x* ∈ *A* : *ax* = 0 for every *a* ∈ *A*} is an ideal (called the *annihilating ideal* of *A*). If *A* is a ring with unity, its annihilating ideal is equal to {0}.

**5** Show that {0} and *A* are ideals of *A*. (They are *trivial* ideals; every other ideal of *A* is a *proper* ideal.) A proper ideal *J* of *A* is called *maximal* if it is not strictly contained in any strictly larger proper ideal: that is, if *J* ⊆ *K*, where *K*is an ideal containing some element not in *J*, then necessarily *K* = *A*. Show that the following is an example of a maximal ideal: In (), the ideal *J* = {*f* : *f*(0) = 0}. [HINT: Use __Exercise D5__. Note that if *g* ∈ *K* and *g*(0) ≠ 0 (that is, *g* ∉ *J*), then the function *h*(*x*) = *g*(*x*) − *g*(*0*) is in *J* hence *h*(*x*) − *g*(*x*) ∈ *K*. Explain why this last function is an invertible element of ().]

**I. Further Properties of Homomorphisms**

Let *A* and *B* be rings. Prove each of the following:

**1** If *f* : *A* → *B* is a homomorphism from *A onto B* with kernel *K*, and *J* is an ideal of *A* such that *K* *J* then *f*(*J*) is an ideal of *B*.

**2** If *f* : *A* → *B* is a homomorphism from *A onto B*, and *B* is a *field*, then the kernel of *f* is a maximal ideal. (HINT: Use part 1, with __Exercise D6__. Maximal ideals are defined in __Exercise H5__.)

**3** There are no nontrivial homomorphisms from to . [The trivial homomorphisms are *f*(*x*) = 0 and *f*(*x*) = *x*.]

**4** If *n* is a multiple of *m*, then * _{m}* is a homomorphic image of

*.*

_{n}**5** If *n* is odd, there is an injective homomorphism from _{2} into _{2n}.

**† J. A Ring of Endomorphisms**

Let *A* be a commutative ring. Prove each of the following:

**1** For each element *a* in *A*, the function *π _{a}* defined by

*π*(

_{a}*x*) =

*ax*satisfies the identity

*π*(

_{a}*x*+

*y*) =

*π*(

_{a}*x*) +

*π*(

_{a}*y*). (In other words,

*π*is an endomorphism of the additive group of

_{a}*A*.)

**2** *π _{a}* is injective iff

*a*is not a divisor of zero. (Assume

*a*≠ 0.)

**3** *π _{a}* is surjective iff

*a*is invertible. (Assume

*A*has a unity.)

**4** Let denote the set {*π _{a}* :

*a*∈

*A*} with the two operations

[*π _{a}* +

*π*](

_{b}*x*) =

*π*(

_{a}*x*) +

*π*(

_{b}*x*) and

*π*=

_{a}π_{b}*π*∘

_{a}*π*

_{b}Verify that is a ring.

**5** If *ϕ* : *A* → is given by *ϕ*(*a*) = *π _{a}*, then

*ϕ*is a homomorphism.

**6** If *A* has a unity, then *ϕ* is an isomorphism. Similarly, if *A* has no divisors of zero then *ϕ* is an isomorphism.