Table 3.3. Equivalent concepts for groups and rings.
groups rings
Abelian commutative
subgroup subring
normal ideal
cyclic principal
Definition 3.21 field A field F is a commutative ring with unity such that, for all a
∈ F, there is an element a
−
1
such that aa
−
1
= a
−
1
a = 1.
In other words, multiplicative inverses exist in fields. A sibling structure of a field is an integral domain, where the elements do not necessarily have multi-
plicative inverses.
Definition 3.22 integral domain An integral domain D is a commutative ring with unity such that, for all nonzero a
, b ∈ D, ab = 0. An integral domain captures the properties of the set of integers in abstract
algebra, hence the name. Other concepts from the set of integers carry over as well.
Definition 3.23 unit, irreducible An element u of an integral domain D is a unit of D if u has a multiplicative inverse in D. A nonzero element p
∈ D that is not a unit of D is an irreducible of D if in any factorization p
= ab in D either a or b is a unit.
So, the concept of primes in Z is generalized to the concept of irreducibles for any integral domain. Fields and integral domains are very much related.
Theorem 3.17 Every field is an integral domain. Every finite integral domain is a field.
Example 3.8 Z , Q, R, C are all rings under the operations of addition and mul-
tiplication. Z
n
, +, ·
n
is a ring where ·
n
is multiplication modulo n. Z is not a field, because it does not have multiplicative inverses for its elements, but Z
is an integral domain. Q and R are fields, and therefore integral domains. Z
p
is an integral domain if p is prime. As Z
p
is finite, Theorem 3.17 implies that Z
is also a field. If p is not a prime, Z
p
is not an integral domain, as it has nonzero elements that divide zero. For example, 2
·
6
3 = 0 in Z
6
.
Another example of a ring we are familiar with is the set of all polynomials with a single variable.
Definition 3.24 polynomial Let ring R to be commutative with unity. A polynomial f
t with coefficients in R is a formal sum ∑
∞
i =0
a
i
t
i
, where a
i
∈ R and t is the indeterminate. The set of all polynomials f
t over R forms a commutative ring R
[t] with unity. For rings, there exists an analog to cyclic Abelian groups, all of whose sub-
groups are normal and cyclic.
Definition 3.25 PID An integral domain D is a principal ideal domain PID if every ideal in D is a principal ideal.
Example 3.9 R, Q, Z, Z
p
for p prime are all PIDs. Usually, R [t] is not a PID
for an arbitrary ring R. However, when R is a field, R [t] becomes a PID.
3.3.3 Modules, Vector Spaces, and Gradings
Recall the definition of a free Abelian group, where we used multiplication to denote multiple additions. We may also view multiplication as an additional
external operation. This makes a free Abelian group a Z-module, as we mul- tiply elements from the group by elements from the ring of integers. Indeed,
any Abelian group is a Z-module following this view.
Definition 3.26 module Let R be a ring. A left R-module consists of an Abelian group M together with an operation of external multiplication of each
element of M by each element of R on the left such that, for all α, β ∈ M and
r , s ∈ R, the following conditions are satisfied:
a rα ∈ M.
b r α + β = rα + rβ.
c r + sα = rα + sα.
d rsα = rsα.
We shall somewhat incorrectly speak of the R-module M. If R is a ring with unity and 1
α = α for all α ∈ M, then M is a unitary R-module. M is cyclic if there exists
α ∈ M such that M = {rα | r ∈ R}. We may also extend the definition of finitely generated groups to modules,
following Definition 3.16. A module is very much like a vector space, with which we are familiar from high school algebra.
Definition 3.27 vector space Let F be a field and V be an Abelian group. A vector space over F
is a unitary F-module, where V is the associated Abelian group. The elements of F are called scalars and the elements of V are called
vectors . We often refer to V as the vector space.
We briefly quickly recall some familiar properties of vector spaces.
Theorem 3.18 basis, dimension If we can write any vector in a vector space
V as a linear combination of the vectors in a finite linearly independent subset B
= {α
i
| i ∈ I} of V , B forms a basis for V and V is finite-dimensional with dimension
|B|. As for free Abelian groups, the dimension is invariant over the set of bases for
the vector space. Our final new concept for this section is that of gradings. Given a ring, we
may be able to decompose the structure into a direct sum decomposition, such that multiplication has a nice form with respect to this decomposition.
Definition 3.28 graded ring A graded
ring is
a ring
R, +, ⊗ equipped with a direct sum decomposition of Abelian groups R ∼
=
i
R
i
, i ∈ Z,
so that multiplication is defined by bilinear pairings R
n
⊗ R
m
→ R
n +m
. El- ements in a single R
i
are homogeneous and have degree i, deg e = i, for all
e ∈ R
i
. If a module is defined over a graded ring as just defined, we may also seek a
similar decomposition for the module.
Definition 3.29 graded module A graded module M over a graded ring R is a module equipped with a direct sum decomposition, M ∼
=
i
M
i
, i ∈ Z, so that the action of R on M is defined by bilinear pairings R
n
⊗ M
m
→ M
n +m
. Our decomposition may be infinite in size. We will be interested, however, in
those gradings that are bounded from below.
Definition 3.30 non-negatively graded A graded ring module is non- negatively graded
if R
i
= 0 M
i
= 0, respectively for all i 0.
Example 3.10 standard grading Let R [t] be the ring of polynomials with
indeterminate t. We may grade R [t] non-negatively with t
n
= t
n
· R[t], n ≥ 0. This is called the standard grading for R
[t].
