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For example, choosing a basis, a symmetric algebra satisfies the universal property and so is a polynomial ring. To give an example, let S be the ring of all functions from R to itself; the addition and the multiplication are those of functions. Let x be the identity function.
The concept of the Jacobson radical of a ring; that is, the intersection of all right (left) annihilators of simple right (left) modules over a ring, is one example. The fact that the Jacobson radical can be viewed as the intersection of all maximal right (left) ideals in the ring, shows how the internal structure of the ring is reflected by ...
An immediate example of a simple ring is a division ring, where every nonzero element has a multiplicative inverse, for instance, the quaternions. Also, for any n ≥ 1 {\displaystyle n\geq 1} , the algebra of n × n {\displaystyle n\times n} matrices with entries in a division ring is simple.
For example, if p is prime and q(X) is an irreducible polynomial with coefficients in the field with p elements, then the quotient ring [] / (()) is a field of characteristic p. Another example: The field C {\displaystyle \mathbb {C} } of complex numbers contains Z {\displaystyle \mathbb {Z} } , so the characteristic of C {\displaystyle \mathbb ...
By convention, a ring has the multiplicative identity. But some authors do not require a ring to have the multiplicative identity; i.e., for them, a ring is a rng. For a rng R, a left ideal I is a subrng with the additional property that is in I for every and every . (Right and two-sided ideals are defined similarly.)
The epimorphic hull and the complete ring of quotients of a real closed ring are again real closed. The (real) holomorphy ring H(A) of a real closed ring A is again real closed. By definition, H(A) consists of all elements f in A with the property −N ≤ f ≤ N for some natural number N. Applied to the examples above, this means that the ...
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The definition is equivalent to saying that a unital associative R-algebra is a monoid object in R-Mod (the monoidal category of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the category of modules.