0

In the definition of subring, it is no necessary that $1$ is in the subring. For example, consider the ring $G=\mathbb{Z}_{10}$ and subring $H=2\mathbb{Z}_{10}.$ In this case, $6$ is the new $1$.

Cristhophel Saramango
  • 11
  • What is your definition of subring? In the one I know, $1$ is required by definition to be in each subring. This implies, for example, that proper ideals are not subrings! – Compacto Feb 15 '22 at 23:03
  • 1
    (If $G$ is commutative) then this can happen iff $G$ is a product of two rings: the 1 of the subring is an element of $G$ such that $e\ne 0,1, e^2=e$ then $G= e G \times (1-e) G$. – reuns Feb 15 '22 at 23:07
  • 3
    @J126 But $3 \not\in 2\mathbf Z_{10}$ – martini Feb 15 '22 at 23:08
  • @J126 6 is the 1 in the subring $2\mathbb{Z_{10}}$ because $6\cdot2k = 12k=2k$ – psl2Z Feb 15 '22 at 23:08
  • @Compacto maybe that it is an additive subgroup and closed under multiplication, so that 1 is not necessary in the subring, so that all ideals are examples of subrings? – psl2Z Feb 15 '22 at 23:10
  • See this post, then: https://math.stackexchange.com/questions/523832/if-a-subring-of-a-ring-r-has-identity-does-r-also-have-the-identity?rq=1 The zero subring (which I would have excluded) has zero as identity. – Compacto Feb 15 '22 at 23:22

1 Answers1

2

This is a language issue - in a technical sense of "language." There are two ways to think of a unital ring:

  • As a set $X$ equipped with a pair of binary operations $\oplus,\odot$ such that [stuff].

  • As a set $X$ equipped with a pair of binary operations $\oplus,\odot$ and a distinguished element (or nullary function) $\mathbb{1}$ such that [stuff].

Call these "weak" and "strong" unital rings, respectively. We actually have a couple more options than this, but this turns out to be a sufficient amount of variety (pun intended - and relevant, see below!) to consider. We then have:

  • If $X$ is a sub-strong-unital-ring of $Y$ then the multiplicative identities of $X$ and $Y$ are the same since that's built right into the definition of sub-thing.

  • If $X$ is merely a sub-weak-unital-ring of $Y$ then the multiplicative identities of $X$ and $Y$ need not coincide (e.g. consider the zero ring as a sub-weak-unital-ring of your favorite nonzero unital ring).

Now typically when we say "subring" in the unital context, we're talking about unital rings in the strong sense and hence we mean the strong subring notion. Note that weak unital rings do not form a variety in the sense of universal algebra - see the link above - so there is some heuristic reason to do this.

Noah Schweber
  • 245,398
  • Though of course in the second sense we have a unary operation and another distinguished element/nullary operation related to the additive structure, though it is not necessary to consider that for the issue at hand. +1 – Arturo Magidin Feb 16 '22 at 03:48