Part 2: Problems, solutions.

Part 3: Problems, solutions.

Part 4: Problems, solutions, homework solutions.

Part 5: Problems, solutions.

### Rings

The idea is that we take as an example the set of integer numbers, and we write out some of their properties. Next, we say that any set along with some defined operations and and chosen elements denoted as and , is a ring (commutative with an identity), if these properties are met. These properties are, for any :

- , ,
- , ,
- ,
- , ,
- for any , there exists (unique, denoted by ), such that .

The integer numbers is obviously an example of a ring, and so is with operations defined modulo . Notice that every field is a ring as well. Indeed, if we add a condition about inverses to the definition of a ring we get the definition of a field!

If is a ring that obviously , i.e. the sets of all polynomials with coefficients in is a ring with operations defined in a natural way.

### Subrings

A subset of a ring (commutative with identity) is called a subring if it includes and and is closed under addition, multiplication and taking negatives. In other words, it is a subring if it is a ring with the same operation as in the bigger ring. E.g. is a subring in . On the other hand, notice that even numbers are not a subring in , because it is closed under addition, multiplication and taking negatives, but does not contain one.

If is a subring of and , we can consider the ring . Then is also a subring of . E.g., is a subring of the complex numbers.

### Properties if elements of a ring, domains

We sat that :

- is a divisor of zero, if there exists , , such that ,
- is a unit, if there exists , że ,
- is a nilpotent, if multiplied a number of times by itself it gives (i.e. for some natural number ).

E.g is a divisor of zero in , because in this ring, . On the other hand, is a unit there, since . Finally, is a nilpotent, because .

Notice, that every nilpotent is a divisor of zero, but no divisor of zero can be a unit (in a non-zero ring).

One can also show that in a finite ring every element is either a unit or a divisor of zero.

A ring is called a integral domain, if it has no divisors of zero. Notice that this means that in a finite integral domain all the non-zero elements are units, and thus every finite integral domain is a field!

Moreover, the degree of a product of two polynomials over a integral domain is not less than the bigger degree of the multiplied polynomials.

### Homomorphisms of rings

Given two rings (commutative with identity) and , a function is a homomorphism if for any : , and . It is easy to check that a homomorphism maps a zero onto zero and a negative onto negative.

E.g. for any integer , given as is a homomorphism which can be easily verified.

An isomorphism of rings is a homomorphism which is a bijection. The kernel of a homomorphism is the set .

### Ideals

A subset of a ring is and ideal, if

- for any , ,
- for any and , .

We are going to denote this fact by .

E.g. even number is an ideal in the ring of integer number, because a sum of even number is even and an even number multiplied by any number is an even number.

The whole ring and the zero ideal are the most trivial examples of ideals. An ideal is proper, if . Notice that an ideal is proper if and only if it does not contain . An ideal is prime if for any , if , then or . It is maximal if there is no ideal , such that . Soon we are going to get to corollary that in the case of finite rings, prime and maximal ideals are the same thing.

Notice that the image of a ring under a homomorphism in a subring, and the image of an ideal is its ideal. Morover, a counterimage of an ideal is an ideal as well. Finally, a kernel of a homeomorhpism is an ideal.

Finally, notice that the only ideals in a field and the zero ideal and the whole field.

### Factorization, domains with unique factorization

Assume now that is an integral domain (so there are no non-zero divisors of zero). A non-zero element is reducible, if it can be written as , where and are not units. If a non-zero element is not a unit and is not reducible it is called irreducible.

E.g. it is clear that is irreducible in . But is reducible. Similarly is irreducible in , but is not.

Sometimes an element can be factorized into a product of irreducible elements in more then one way. E.g. in . We shall say that an element has a unique factorization, if whenever , where are irreducible, we have and elements and can be set into pairs such that for a unit .

Integral domains in which all non-zero element have unique factorization are called domains with unique factorization. An example of such a domain are the integrals themselves, or e.g. .

### Generated ideals and quotient rings

The ideal generated by is the least ideal , such that . We denote it by . Notice that . If is a one-element set, then is called a principal ideal and we write simply . Obviously, then .

It is easy to prove that is a field if and only if it is non-zero and the only ideals in it are and .

If is an ideal in and , we define a coset . Notice that are in the same coset of is and only if . The set of all cosets is denoted by and we can introduce a structure of a ring in it stipulating that zero is , identity is , and , and . Such a ring is called the quotient ring.

E.g. take the ideal of even numbers in . Then is isomorphic to . Indeed, two numbers are in the same coset of the ideal if and only if both are even or both are odd, so there are only two cosets, and the operations on these cosets are consistent with operations modulo . Thus defined as is an isomorphism of rings.

One can also prove the following important theorem, called the theorem about a homomorphism. If is a homomorphism, then there exists exactly one homomorphism such that , where is given by . Moreover, is an isomorphism of and and thus we have a one-to-one correspondence between ideals in the ring and ideals of containing .

This leads to the following characterization:

- is a field if and only if the ideal is maximal,
- is an integral domain if and only if the ideal is prime.

In particular, every maximal ideal is prime. By Zorn Lemma, every proper ideal is contained in a maximal ideal. Thus, every ring can be mapped onto a field.

### Divisibility, domains of proper ideals

In a domain , , if there exists such that . If and , then we say that is associated to , and write . It is easy to prove that if and only if . Moreover, if and only if there exists a unit such that .

An element is a prime, if and only if for any such that , or . It is clear that is a prime if and only if is a prime ideal, and that every prime ideal is irreducible. This implication can be reversed in domains with unique factorization. In such domains every irreducible element is a prime. Actually, if in a domain every element which is nonzero and is not a unit can be factorized into irreducible elements, and every irreducible element is a prime, then it is a domain with unique factorization.

In a domain , an element is a greatest common divisor of (we write ), if , , and for every such that and , also . In general, the greatest common divisor may not exist. It is clear that it always exists in a domain with unique factorization.

A ring is called a ring of principal ideals, if every ideal in it is principal. A domain which is a ring of principal ideals is called a domain of principal ideals. Notice that and for any field are domains of principal ideals, but is not.

Every ideal generated by an irreducible element in a domain of principal ideals is maximal, thus every prime ideal in such a domain is either zero ideal or maximal. From this it follows that every domain of principal ideals is a domain with unique factorization.

Moreover, in a domain of principal ideal it is easy to see that every two elements have their gcd. Indeed, if and only if .

### Euclidean domains

There is this known algorithm of finding the gcd of two integers. If we want to find (assume, that , ) we find such that , where and then, if , then , so we can repeat the step for smaller numbers. If, on the other hand , the divisor we are looking for is . E.g. we can find in the following way , , , thus .

This algorithm ends because in every step the rest in in some sense smaller — in this case in the sense of absolute value. We may use this algorithm in other rings as well provided that in the given ring we find a property (called a norm) according to which the rest descreases in every step, because only this guarantees that the algorithm eventually stops.

An norm in a domain is a function such that is and only if and . A norm is eauclidean (thus, it allows to use the Euclid algorithm to find gcd), if morover for any non-zero there exist and , such that and . The domains for which there exists an euclidean norm is called euclidean domains. One can prove that every euclidean domani is a domain of principal ideals. Both with the norm being absolute value, and for any field with the norm being the degree of a polynomial are euclidean domains. There exists principal ideal domains which are not euclidean, but they are not very easy to find, e.g. .

It is worth noticing that every field is an euclidean domain in a not very interesting and quite trivial way — with the norm equal to for and for any other element.