IGNOU BMTE 144 2026 - Which of the following statements are true, and which are false? Give

Question

Which of the following statements are true, and which are false? Give reasons for your answers.
i) For any ring R and $a, b \in R, (a + b)^2 = a^2 + 2ab + b^2$ .
ii) Every ring has at least two elements.
iii) If R is a ring with identity and I is an ideal of R, then the identity of R/I is the same as the identity of R.
iv) If $f : R \rightarrow S$ is a ring homomorphism, then it is a group homomorphism from (R, +) to (S, +).
v) If R is a ring, then any ring homomorphism from $R \times R$ into R is surjective.

10 Jan 2026

Answer :

Word Count : 171

i) False – Not every equation over a ring R has a solution. For example, in the ring of integers ℤ, the equation 2x = 1 has no solution in ℤ. ii) False – A ring can have only one element, called the zero ring, where 0 = 1. For example, the set {0} with usual addition and multiplication forms

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