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A (graded) [[Ring|ring]] object in the [[Category|category]] of (graded) co-commutative co-algebras (cf. [[Co-algebra|Co-algebra]]). Such an object consists, first, of a sequence  $  \{ H _ {i} \} $
of [[Abelian group|Abelian group]] objects in the category. These are better known as commutative Hopf algebras with conjugation. Since they belong to the category, they have a coproduct:

$$ 
\psi : {H _ {i} } \rightarrow {H _ {i} \otimes H _ {i} } .
$$

Let  $  \psi ( x ) = \sum x  ^  \prime  \otimes x ^ {\prime \prime } $. 
As group objects in a category of co-algebras, the  $  H _ {i} $
also have a product  $  H _ {i} \otimes H _ {i} \rightarrow H _ {i} $. 
Below, this product is denoted by  $  * $. 
The  "*"  product should be thought of as  "addition"  in the ring as it is the pairing which gives the Abelian group structure. For ring  "multiplication"  one has

$$ 
\circ : {H _ {i} \otimes H _ {j} } \rightarrow {H _ {i + j }  } .
$$

As with any ring, there must be a distributive law relating the multiplication and the addition. Chasing diagrams in the category one sees that it is:

$$ 
x \circ ( y * z ) = \sum \pm  ( x  ^  \prime  \circ y ) * ( x  ^  \prime  \circ z ) .
$$

Hopf rings arise naturally in the study of the  $  \Omega $-
spectra associated with generalized cohomology theories. Any generalized [[Cohomology|cohomology]] theory,  $  G  ^ {*} ( X ) $, 
gives rise to a sequence of spaces,  $  \{ {\underline{G} } _ {k} \} $, 
with the property that  $  G  ^ {k} ( X ) \simeq [ X, {\underline{G} } _ {k} ] $, 
the homotopy classes of mappings. If  $  G $
is a multiplicative theory, then  $  \{ {\underline{G} } _ {k} \} $
is a graded ring object in the homotopy category. If  $  E $
represents a generalized homology theory and if there is a Künneth isomorphism for the  $  E $-
homology of the spaces in the  $  \Omega $-
spectra for  $  G $, 
then the sequence  $  \{ E _ {*} ( {\underline{G} } _ {*} ) \} $
becomes a Hopf ring. One can thus use knowledge of generalized homologies to further the understanding of generalized cohomologies by studying their classifying spaces using Hopf rings.

There are a number of Hopf rings which have been computed. Examples are  $  E _ {*} ( {\underline{BP } } _ {*} ) $
and  $  E _ {*} ( {\underline{MU } } _ {*} ) $, 
$  E $
a complex orientable theory, [[#References|[a9]]] (the basic reference for Hopf rings);  $  E _ {*} ( {\underline{K ( n ) } } _ {*} ) $
and  $  E _ {*} ( {\underline{P ( n ) } } _ {*} ) $, 
$  E $
a complex orientable theory with  $  I _ {n} = 0 $, 
[[#References|[a14]]] and [[#References|[a8]]];  $  H _ {*} ( K ( \mathbf Z/ {( p ) } , * ) ) $, 
[[#References|[a13]]], § 8;  $  K ( n ) _ {*} ( - ) $
for Eilenberg–MacLane spaces, [[#References|[a10]]];  $  K ( n ) _ {*} ( {\underline{k ( n ) } } _ {\ * }  ) $, 
[[#References|[a5]]];  $  H _ {*} ( {\underline{KO } } ) $, 
[[#References|[a11]]]; and the breakthrough description of  $  H _ {*} ( QS  ^ {0} , \mathbf Z/ ( 2 ) ) $
in [[#References|[a12]]], and its sequel for  $  H _ {*} ( QS  ^ {*} , \mathbf Z/ ( 2 ) ) $
in [[#References|[a1]]] followed by corresponding results for odd primes in [[#References|[a7]]]. Other references are [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], and [[#References|[a6]]].

Hopf rings have a very rich algebraic structure, useful in two distinct ways: descriptive and computational. All of the above examples have their Hopf rings described with just a few generators and relations. The computations are generally carried out using Hopf ring techniques as well.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.J. Eccles,   P.R. Turner,   W.S. Wilson,   "On the Hopf ring for the sphere"  ''Math. Z.'' , '''224 (2)'''  (1997)  pp. 229–233</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.J. Hopkins,   J.R. Hunton,   "The structure of spaces representing a Landweber exact cohomology theory"  ''Topology'' , '''34 (1)'''  (1995)  pp. 29–36</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.R. Hunton,   N. Ray,   "A rational approach to Hopf rings"  ''J. Pure Appl. Algebra'' , '''101 (3)'''  (1995)  pp. 313–333</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Kashiwabara,   "Hopf rings and unstable operations"  ''J. Pure Appl. Algebra'' , '''194'''  (1994)  pp. 183–193</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Kramer,   "The periodic Hopf ring of connective Morava <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110300/h11030033.png" />-theory"  ''Ph.D. Thesis, Johns Hopkins Univ.''  (1990)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  T. Kashiwabara,   N.P. Strickland,   P.R. Turner,   "Morava <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110300/h11030034.png" />-theory Hopf ring for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110300/h11030035.png" />"  C. Broto (ed.)  etAAsal. (ed.) , ''Algebraic Topology: New Trends in Localization and Periodicity'' , ''Progress in Mathematics'' , '''139''' , Birkhäuser  (1996)  pp. 209–222</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  Y. Li,   "On the Hopf ring for the sphere"  ''Ph.D. Thesis, Johns Hopkins Univ.''  (1996)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  D.C. Ravenel,   W.S. Wilson,   "The Hopf ring for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110300/h11030036.png" />"  ''Canadian J. Math.'' , '''48 (5)'''  (1996)  pp. 1044–1063</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  D.C. Ravenel,   W.S. Wilson,   "The Hopf ring for complex cobordism"  ''J. Pure Appl. Algebra'' , '''9'''  (1977)  pp. 241–280</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  D.C. Ravenel,   W.S. Wilson,   "The Morava <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110300/h11030037.png" />-theories of Eilenberg–Mac Lane spaces and the Conner–Floyd conjecture"  ''Amer. J. Math.'' , '''102'''  (1980)  pp. 691–748</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  N. Strickland,   "Bott periodicity and Hopf rings"  ''Ph.D. Thesis, Univ. Manchester''  (1992)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  P.R. Turner,   "Dickson coinvariants and the homology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110300/h11030038.png" />"  ''Math. Z.'' , '''224 (2)'''  (1997)  pp. 209–228</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  W.S. Wilson,   "Brown–Peterson homology: an introduction and sampler" , ''CBMS'' , '''48''' , Amer. Math. Soc.  (1982)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  W.S. Wilson,   "The Hopf ring for Morava <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110300/h11030039.png" />-theory"  ''Publ. RIMS Kyoto Univ.'' , '''20'''  (1984)  pp. 1025–1036</TD></TR></table>