Almost strongly unital rings (Q6174693)

As the authors mentioned, if \(P\) is a certain property, then a mathematical structure \(\mathcal{A}\) almost has property \(P\) if \(\mathcal{A}\) does not have property \(P\), but every substructure (or quotient structure) of \(\mathcal{A}\) has property \(P\). If \(R\) is a ring (not necessarily commutative or with identity), \(S\subseteq R\) is called a subring of \(R\), if \((S,+)\) is a subgroup of \((R,+)\) and \(S\) is closed under the multiplication of \(R\). If each subring \(S\) of \(R\) has an identity, say \(1_S\) (it is possible that \(1_R\neq 1_S\)), then \(R\) is called strongly unital ring. These rings completely determined in [\textit{G. Oman} and \textit{J. Stroud}, Involve 13, No. 5, 823--828 (2020; Zbl 1479.16002)]. In fact, they proved that a nontrivial ring \(R\) is strongly unital if and only if \(R\cong F_1\times\cdots\times F_n\), where each \(F_i\) is an absolutely algebraic field (i.e., field with nonzero characteristic which is algebraic over its prime subfield). In this article, a ring \(R\) is called almost strongly unital, if \(R\) has no identity but every proper subring of \(R\) has an identity. In Lemma 6, they prove that each commutative reduced Artinian ring has an identity. In Theorem 8, they prove that if \(R\) is a nonzero ring, then every proper subring of \(R\) has an identity if and only if either \(R\) is strongly unital or \(R\cong \frac{X\mathbb{F}_p[X]}{<X^2>}\), where \(p\) is a prime number and \(\mathbb{F}_p\) is a field with exactly \(p\) elements.