91Ó°ÊÓ

Compute the product in the given ring. $$ (20)(-8) \text { in } Z_{26} $$

Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field. $$ \mathbb{Z} \times \mathbb{Z} \text { with addition and multiplication by components } $$

In Exercises 14 through 19 , describe all units in the given ring \(\mathrm{Z}_{5}\)

Describe all units in the given ring. $$ z_{5} $$

Consider the matrix ring \(M_{2}\left(Z_{2}\right)\). a. Find the order of the ring, that is, the number of elements in it. \(\mathbf{h}\) List all units in the ring.

(Linear algebra) Consider the map det of \(M_{n}(\mathbb{R})\) into \(\mathbb{R}\) where \(\operatorname{det}(A)\) is the determinant of the matrix \(A\) for \(A \in M_{n}(\mathbb{R})\). Is det a ring homomorphism? Why or why not?

Describe all ring homomorphisms of \(Z\) into \(Z\).

Give an example of a ring with unity \(1 \neq 0\) that has a subring with nonzero unity \(1^{\prime} \neq 1\). [Hlm: Consider a direct product, or a subring of \(Z_{6 .}\).]

Mark each of the following true or false. a. Every field is also a ring. b. Every ring has a multiplicative identity. c. Every ring with unity has at least two units. d. Every ring with unity has at most two units. e. It is possible for a subset of some field to be a ring but not a subfield, under the induced operations. f. The distributive laws for a ring are not very important. g. Multiplication in a field is commutative. h. The nonzero elements of a field form a group under the multiplication in the field, i. Addition in every ring is commutative. j. Every element in a ring has an additive inverse.

Let \((R,+)\) be an abelian group. Show that \(\left(R,+,^{\circ}\right)\) is a ring if we define \(a b=0\) for all \(a, b \in R\).