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. $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
i) If k is a field, then so is $k \times k$ .
ii) If R is an integral domain and I is an ideal of R, then $\text{Char}(R) = \text{Char}(R/I)$ .
iii) In a domain, every prime ideal is a maximal ideal.
iv) If R is a ring with zero divisors, and S is a subring of R, then S has zero divisors.
v) If R is a ring and $f(x) \in R[x]$ is of degree $n \in \mathbb{N}$ , then f(x) has exactly n roots in R.

10 Jan 2026

Answer :

Word Count : 225

i) False. Just because (k) is a field does not automatically make an equation a field. An equation is a relation, not a set with field operations. Fields are defined as sets with addition and multiplication satisfying certain axioms, so the statement is incorrect. ii) False. If (R) is an integral domain and (I)

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