ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

10. 2: Subring
117. Definition: Delring
A substructure of a ring which is a ring in its own right is said to be a subring.
118. Theorem: Any substructure of ring is a subring
Let D be a substructure of the ring R. Then D is a subring.
Proof : When D is a substructure as a ring then it is also a substructure of
the underlying additive group and so (D, +, 0, −) is itself a group, obviously
commutative. In a similar way we see that (D, ∗, 1) is a monoid. Distributivity
follows since this propery is hereditary.
This may be formulated as the following criterion
119. Theorem: Subring criterion
A subset D of a ring (R, +, −, 0, ∗, 1) is a subring if and only if D + D ⊆
D,D ∗ D ⊆ D,−D ⊆ D and 1 ∈ D.
Proof : The conditions are clearly necessary. And they explicitly guaranties
the closure of all the operations except 0. To see that 0 =∈ D notice that
0 = 1 − 1.
10. 3: Induced rings
120. Theorem: Function spaces
Suppose that A is a ring and that M is a set. The algebraic structure on
F(M, A) induced by A is a ring.
Proof : (F(M, A), +, −, 0) is a commutative group since it is induced by the
underlying additive group. In the same way we see that (F(M, A), ∗, 1) is a
monoid. Distributivity is inherited.
The preceding theorem furnishes many important examples of rings, the rings
of functions: X56, with some important subrings, rings of polynomials E57.
121. Theorem: The product of rings is a ring
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