is this a subring? if $I = R$ x $S$ is a ring (which implies that $S$ and $R$ are rings), can we say that $S$ and $R$ are subrings of $I$?
Also how come wikipedia and many textbooks define a subring as having the identity $1_{R}$ as an element to be a requirement while for instance $2 \mathbb{Z} \subset \mathbb{Z}$ has no $1_{\mathbb{Z}}$ !?
 A: If you don't require ring homs to map $1$ to $1$ (hence same for inclusion homs, i.e. subrings have the same $1$) then lots of things can go wrong. For example, consider
Lemma $ $If $S$ is a ring, and $x,y\in S$ are coprime in the sense that
there exist $a$ and $b$ in $S$ such that  $a x + b y = 1,$ then for any ring
$ R$ that contains $S,$ $x,y$ remain coprime in $R$ (in the same sense).
Counterexample $\ $  Let  $\ S = \Bbb Z \times\! \{0\} \subset R = \Bbb Z \times \Bbb Z,\,$ and $\,  a = (3,0),\ b = (2,0)$  
Then  $\,aS + bS = S = aR + bR \ne R,\,$ so  $a,b$  are coprime in $S$ but not in $R.$
The lemma requires  $\,S \subset R\,$  is a sub(ring with $1),$ so $\, 1_S = 1_R.$
However, in some contexts non-unital rings are more natural, e.g. they are employed heavily in the general study of radical theories for rings. Perhaps you will find the following remarks of interest, excerpted from the preface of Gardner and Wiegandt: Radical Theory of Rings, 2004.

Some authors deal exclusively with rings with unity element. This assumption is all right and not restrictive, if the ring is ﬁxed, as in module theory or group ring theory or sometimes investigating polynomial rings and power series rings (if the ring of coefficients does not possess a unity element. the indeterminate x is not a member of the polynomial ring). Dealing, however, simultaneously with several objects in a category of rings, demanding the existence of a unity element leads to a bizarre situation. Rings with unity element include among their fundamental operations the nullary operation $\mapsto$ 1 assigning the unity element. Thus in the category of rings with unity element the morphisms, in particular the monomorphisms, have to preserve also this nullary operation: subrings (i.e. subobjects) have to contain the same unity element, and so a proper ideal with unity element is not a subring, although a ring and a direct summand; there are no inﬁnite direct sums, no nil rings, no Jacobson radical rings, the ﬁnite valued linear transformations of an inﬁnite dimensional vector space do not form a ring, etc. Thus, in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable. This applies also to radical theory. and so in this book rings need not have a unity element.

A: For whatever reason, terminology in ring theory is not as standardized as terminology in group theory. This is a constant (but minor) nuisance.
The rings that I work with, and the rings in all neighboring branches of math, are commutative, with 1. Therefore, one would want to define a substructure to also contain $1$ and in particular $S$ is not a subring of $S \oplus R$ (in general). I would prefer it if everyone always meant this when they said "ring, subring, etc." and found a different word for not-necessarily commutative rings (that have 1) and a still different word for (rings that don't have to have $1$).
However, I believe there are folks who work with rings without $1$, and I am sure that in their areas of math, it would be annoying and silly to keep saying ("thing that is like a ring except it need not have $1$"), so from their perspectives, maybe things are different. In this case, clearly it's silly to require a subring to contain $1$ (even if the ambient ring happens to have it) and there is a natural ring map $R \to R \oplus S$ ($x \mapsto (x, 0)$).
(The general philosophy uniting these answers is, if we are studying the theory of sets with certain "special elements" (like $0, 1$) and "operations" (like $+, \times$), then a "sub-(whatever)" should not just be a subset which happens to be a (whatever) but a subset which is a whatever w.r.t. to the same special elements and operations as the ambient set. And the reason there are two different answers is that people disagree about what "ring" means.)
