Abstract Algebra: Rings and Fields

A ring is a non-empty set R equipped with two binary operations, addition (+) and multiplication (x), that satisfy the following properties:

1. Associativity of addition: For all a, b, c in R, (a + b) + c = a + (b + c).
2. Commutativity of addition: For all a, b in R, a + b = b + a.
3. Existence of additive identity: There exists an element 0 in R such that for all a in R, a + 0 = a.
4. Existence of additive inverse: For each a in R, there exists an element -a in R such that a + (-a) = 0.
5. Associativity of multiplication: For all a, b, c in R, (a x b) x c = a x (b x c).
6. Distributivity of multiplication over addition: For all a, b, c in R, a x (b + c) = (a x b) + (a x c).

The set of integers (Z) forms a ring under the usual addition and multiplication operations because it satisfies all the properties of a ring. The other options are not rings because they do not satisfy all the properties of a ring.

A commutative ring is a ring in which the multiplication operation is commutative, meaning that for all a, b in R, a x b = b x a. The other options are not correct because they do not accurately define a commutative ring.

The set of integers (Z) forms a commutative ring under the usual addition and multiplication operations because the multiplication operation is commutative, meaning that for all a, b in Z, a x b = b x a.

A division ring is a ring in which every nonzero element has a multiplicative inverse, meaning that for every nonzero element a in R, there exists an element b in R such that a x b = b x a = 1, where 1 is the multiplicative identity of R. The other options are not correct because they do not accurately define a division ring.

The set of rational numbers (Q) forms a division ring under the usual addition and multiplication operations because every nonzero rational number has a multiplicative inverse. The other options are not division rings because they do not satisfy this property.

A field is a commutative ring in which every nonzero element has a multiplicative inverse. This means that for every nonzero element a in F, there exists an element b in F such that a x b = b x a = 1, where 1 is the multiplicative identity of F. The other options are not correct because they do not accurately define a field.

The set of rational numbers (Q) forms a field under the usual addition and multiplication operations because it is a commutative ring and every nonzero rational number has a multiplicative inverse. The other options are not fields because they do not satisfy this property.

The zero element of a ring is the element that, when added to any other element in the ring, leaves that element unchanged. It is usually denoted by 0 or +0. The other options are not correct because they do not accurately define the zero element of a ring.

The multiplicative identity of a ring is the element that, when multiplied by any other element in the ring, leaves that element unchanged. It is usually denoted by 1 or x1. The other options are not correct because they do not accurately define the multiplicative identity of a ring.

The additive inverse of an element a in a ring is the element -a such that a + (-a) = 0, where 0 is the zero element of the ring. The other options are not correct because they do not accurately define the additive inverse of an element in a ring.

The multiplicative inverse of an element a in a division ring is the element b such that a x b = b x a = 1, where 1 is the multiplicative identity of the division ring. The other options are not correct because they do not accurately define the multiplicative inverse of an element in a division ring.

An ideal of a ring R is a non-empty subset I of R that is closed under addition and multiplication, meaning that for all a, b in I and r in R, a + b is in I and r x a is in I. The other options are not correct because they do not accurately define an ideal of a ring.

The set of even integers is an ideal of the ring of integers (Z) because it is a non-empty subset of Z that is closed under addition and multiplication. The other options are not ideals of Z because they are not closed under addition or multiplication.

A maximal ideal of a ring R is an ideal M that is not properly contained in any other ideal of R. This means that there is no ideal I in R such that M is a proper subset of I. The other options are not correct because they do not accurately define a maximal ideal of a ring.