Let R is commutative.
We are given: * $R$ is a commutative ring. * For all $a \in R$, we are told that $a^2 = a$. * We are to show that: $a = -a$, i.e., every element is its own additive inverse. --- ### Step-by-step proof: Given: $$ a^2 = ____ ________ __________ ________ _______ ____.
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Let R is commutative.
State Lagrange’s theorem. What are the possible orders of subgroups of a group of order 12?
as a product of transpositons. What is the signature of?
Let R be a ring in which a² = a for all a ∈ R. Show that a = -a and R is commutative.
व्यवसाय योजना क्या है? व्यवसाय योजना की व्यवहार्यता निर्धारित करने के लिए किए जाने वाले विभिन्न प्रकार के विश्लेषणों पर चर्चा करें
Define an integral domain. Give an example of an integral domain which is not a field.
Which of the following statements are true and which are false? Justify your answer with a short proof or a counter example.
(a) Every subgroup of S3 is normal.
(b) Every abelian group is cyclic.
(c) In a ring with unity, the sum of any two units is a unit.
(d) If a field has characteristic p, p a prime, the field is finite.
(e) If every element in group has finite order, the group is finite.